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.venv/lib/python3.13/site-packages/sympy/concrete/tests/test_delta.py

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+from sympy.concrete import Sum
+from sympy.concrete.delta import deltaproduct as dp, deltasummation as ds, _extract_delta
+from sympy.core import Eq, S, symbols, oo
+from sympy.functions import KroneckerDelta as KD, Piecewise, piecewise_fold
+from sympy.logic import And
+from sympy.testing.pytest import raises
+
+i, j, k, l, m = symbols("i j k l m", integer=True, finite=True)
+x, y = symbols("x y", commutative=False)
+
+
+def test_deltaproduct_trivial():
+    assert dp(x, (j, 1, 0)) == 1
+    assert dp(x, (j, 1, 3)) == x**3
+    assert dp(x + y, (j, 1, 3)) == (x + y)**3
+    assert dp(x*y, (j, 1, 3)) == (x*y)**3
+    assert dp(KD(i, j), (k, 1, 3)) == KD(i, j)
+    assert dp(x*KD(i, j), (k, 1, 3)) == x**3*KD(i, j)
+    assert dp(x*y*KD(i, j), (k, 1, 3)) == (x*y)**3*KD(i, j)
+
+
+def test_deltaproduct_basic():
+    assert dp(KD(i, j), (j, 1, 3)) == 0
+    assert dp(KD(i, j), (j, 1, 1)) == KD(i, 1)
+    assert dp(KD(i, j), (j, 2, 2)) == KD(i, 2)
+    assert dp(KD(i, j), (j, 3, 3)) == KD(i, 3)
+    assert dp(KD(i, j), (j, 1, k)) == KD(i, 1)*KD(k, 1) + KD(k, 0)
+    assert dp(KD(i, j), (j, k, 3)) == KD(i, 3)*KD(k, 3) + KD(k, 4)
+    assert dp(KD(i, j), (j, k, l)) == KD(i, l)*KD(k, l) + KD(k, l + 1)
+
+
+def test_deltaproduct_mul_x_kd():
+    assert dp(x*KD(i, j), (j, 1, 3)) == 0
+    assert dp(x*KD(i, j), (j, 1, 1)) == x*KD(i, 1)
+    assert dp(x*KD(i, j), (j, 2, 2)) == x*KD(i, 2)
+    assert dp(x*KD(i, j), (j, 3, 3)) == x*KD(i, 3)
+    assert dp(x*KD(i, j), (j, 1, k)) == x*KD(i, 1)*KD(k, 1) + KD(k, 0)
+    assert dp(x*KD(i, j), (j, k, 3)) == x*KD(i, 3)*KD(k, 3) + KD(k, 4)
+    assert dp(x*KD(i, j), (j, k, l)) == x*KD(i, l)*KD(k, l) + KD(k, l + 1)
+
+
+def test_deltaproduct_mul_add_x_y_kd():
+    assert dp((x + y)*KD(i, j), (j, 1, 3)) == 0
+    assert dp((x + y)*KD(i, j), (j, 1, 1)) == (x + y)*KD(i, 1)
+    assert dp((x + y)*KD(i, j), (j, 2, 2)) == (x + y)*KD(i, 2)
+    assert dp((x + y)*KD(i, j), (j, 3, 3)) == (x + y)*KD(i, 3)
+    assert dp((x + y)*KD(i, j), (j, 1, k)) == \
+        (x + y)*KD(i, 1)*KD(k, 1) + KD(k, 0)
+    assert dp((x + y)*KD(i, j), (j, k, 3)) == \
+        (x + y)*KD(i, 3)*KD(k, 3) + KD(k, 4)
+    assert dp((x + y)*KD(i, j), (j, k, l)) == \
+        (x + y)*KD(i, l)*KD(k, l) + KD(k, l + 1)
+
+
+def test_deltaproduct_add_kd_kd():
+    assert dp(KD(i, k) + KD(j, k), (k, 1, 3)) == 0
+    assert dp(KD(i, k) + KD(j, k), (k, 1, 1)) == KD(i, 1) + KD(j, 1)
+    assert dp(KD(i, k) + KD(j, k), (k, 2, 2)) == KD(i, 2) + KD(j, 2)
+    assert dp(KD(i, k) + KD(j, k), (k, 3, 3)) == KD(i, 3) + KD(j, 3)
+    assert dp(KD(i, k) + KD(j, k), (k, 1, l)) == KD(l, 0) + \
+        KD(i, 1)*KD(l, 1) + KD(j, 1)*KD(l, 1) + \
+        KD(i, 1)*KD(j, 2)*KD(l, 2) + KD(j, 1)*KD(i, 2)*KD(l, 2)
+    assert dp(KD(i, k) + KD(j, k), (k, l, 3)) == KD(l, 4) + \
+        KD(i, 3)*KD(l, 3) + KD(j, 3)*KD(l, 3) + \
+        KD(i, 2)*KD(j, 3)*KD(l, 2) + KD(i, 3)*KD(j, 2)*KD(l, 2)
+    assert dp(KD(i, k) + KD(j, k), (k, l, m)) == KD(l, m + 1) + \
+        KD(i, m)*KD(l, m) + KD(j, m)*KD(l, m) + \
+        KD(i, m)*KD(j, m - 1)*KD(l, m - 1) + KD(i, m - 1)*KD(j, m)*KD(l, m - 1)
+
+
+def test_deltaproduct_mul_x_add_kd_kd():
+    assert dp(x*(KD(i, k) + KD(j, k)), (k, 1, 3)) == 0
+    assert dp(x*(KD(i, k) + KD(j, k)), (k, 1, 1)) == x*(KD(i, 1) + KD(j, 1))
+    assert dp(x*(KD(i, k) + KD(j, k)), (k, 2, 2)) == x*(KD(i, 2) + KD(j, 2))
+    assert dp(x*(KD(i, k) + KD(j, k)), (k, 3, 3)) == x*(KD(i, 3) + KD(j, 3))
+    assert dp(x*(KD(i, k) + KD(j, k)), (k, 1, l)) == KD(l, 0) + \
+        x*KD(i, 1)*KD(l, 1) + x*KD(j, 1)*KD(l, 1) + \
+        x**2*KD(i, 1)*KD(j, 2)*KD(l, 2) + x**2*KD(j, 1)*KD(i, 2)*KD(l, 2)
+    assert dp(x*(KD(i, k) + KD(j, k)), (k, l, 3)) == KD(l, 4) + \
+        x*KD(i, 3)*KD(l, 3) + x*KD(j, 3)*KD(l, 3) + \
+        x**2*KD(i, 2)*KD(j, 3)*KD(l, 2) + x**2*KD(i, 3)*KD(j, 2)*KD(l, 2)
+    assert dp(x*(KD(i, k) + KD(j, k)), (k, l, m)) == KD(l, m + 1) + \
+        x*KD(i, m)*KD(l, m) + x*KD(j, m)*KD(l, m) + \
+        x**2*KD(i, m - 1)*KD(j, m)*KD(l, m - 1) + \
+        x**2*KD(i, m)*KD(j, m - 1)*KD(l, m - 1)
+
+
+def test_deltaproduct_mul_add_x_y_add_kd_kd():
+    assert dp((x + y)*(KD(i, k) + KD(j, k)), (k, 1, 3)) == 0
+    assert dp((x + y)*(KD(i, k) + KD(j, k)), (k, 1, 1)) == \
+        (x + y)*(KD(i, 1) + KD(j, 1))
+    assert dp((x + y)*(KD(i, k) + KD(j, k)), (k, 2, 2)) == \
+        (x + y)*(KD(i, 2) + KD(j, 2))
+    assert dp((x + y)*(KD(i, k) + KD(j, k)), (k, 3, 3)) == \
+        (x + y)*(KD(i, 3) + KD(j, 3))
+    assert dp((x + y)*(KD(i, k) + KD(j, k)), (k, 1, l)) == KD(l, 0) + \
+        (x + y)*KD(i, 1)*KD(l, 1) + (x + y)*KD(j, 1)*KD(l, 1) + \
+        (x + y)**2*KD(i, 1)*KD(j, 2)*KD(l, 2) + \
+        (x + y)**2*KD(j, 1)*KD(i, 2)*KD(l, 2)
+    assert dp((x + y)*(KD(i, k) + KD(j, k)), (k, l, 3)) == KD(l, 4) + \
+        (x + y)*KD(i, 3)*KD(l, 3) + (x + y)*KD(j, 3)*KD(l, 3) + \
+        (x + y)**2*KD(i, 2)*KD(j, 3)*KD(l, 2) + \
+        (x + y)**2*KD(i, 3)*KD(j, 2)*KD(l, 2)
+    assert dp((x + y)*(KD(i, k) + KD(j, k)), (k, l, m)) == KD(l, m + 1) + \
+        (x + y)*KD(i, m)*KD(l, m) + (x + y)*KD(j, m)*KD(l, m) + \
+        (x + y)**2*KD(i, m - 1)*KD(j, m)*KD(l, m - 1) + \
+        (x + y)**2*KD(i, m)*KD(j, m - 1)*KD(l, m - 1)
+
+
+def test_deltaproduct_add_mul_x_y_mul_x_kd():
+    assert dp(x*y + x*KD(i, j), (j, 1, 3)) == (x*y)**3 + \
+        x*(x*y)**2*KD(i, 1) + (x*y)*x*(x*y)*KD(i, 2) + (x*y)**2*x*KD(i, 3)
+    assert dp(x*y + x*KD(i, j), (j, 1, 1)) == x*y + x*KD(i, 1)
+    assert dp(x*y + x*KD(i, j), (j, 2, 2)) == x*y + x*KD(i, 2)
+    assert dp(x*y + x*KD(i, j), (j, 3, 3)) == x*y + x*KD(i, 3)
+    assert dp(x*y + x*KD(i, j), (j, 1, k)) == \
+        (x*y)**k + Piecewise(
+            ((x*y)**(i - 1)*x*(x*y)**(k - i), And(1 <= i, i <= k)),
+            (0, True)
+        )
+    assert dp(x*y + x*KD(i, j), (j, k, 3)) == \
+        (x*y)**(-k + 4) + Piecewise(
+            ((x*y)**(i - k)*x*(x*y)**(3 - i), And(k <= i, i <= 3)),
+            (0, True)
+        )
+    assert dp(x*y + x*KD(i, j), (j, k, l)) == \
+        (x*y)**(-k + l + 1) + Piecewise(
+            ((x*y)**(i - k)*x*(x*y)**(l - i), And(k <= i, i <= l)),
+            (0, True)
+        )
+
+
+def test_deltaproduct_mul_x_add_y_kd():
+    assert dp(x*(y + KD(i, j)), (j, 1, 3)) == (x*y)**3 + \
+        x*(x*y)**2*KD(i, 1) + (x*y)*x*(x*y)*KD(i, 2) + (x*y)**2*x*KD(i, 3)
+    assert dp(x*(y + KD(i, j)), (j, 1, 1)) == x*(y + KD(i, 1))
+    assert dp(x*(y + KD(i, j)), (j, 2, 2)) == x*(y + KD(i, 2))
+    assert dp(x*(y + KD(i, j)), (j, 3, 3)) == x*(y + KD(i, 3))
+    assert dp(x*(y + KD(i, j)), (j, 1, k)) == \
+        (x*y)**k + Piecewise(
+            ((x*y)**(i - 1)*x*(x*y)**(k - i), And(1 <= i, i <= k)),
+            (0, True)
+        ).expand()
+    assert dp(x*(y + KD(i, j)), (j, k, 3)) == \
+        ((x*y)**(-k + 4) + Piecewise(
+            ((x*y)**(i - k)*x*(x*y)**(3 - i), And(k <= i, i <= 3)),
+            (0, True)
+        )).expand()
+    assert dp(x*(y + KD(i, j)), (j, k, l)) == \
+        ((x*y)**(-k + l + 1) + Piecewise(
+            ((x*y)**(i - k)*x*(x*y)**(l - i), And(k <= i, i <= l)),
+            (0, True)
+        )).expand()
+
+
+def test_deltaproduct_mul_x_add_y_twokd():
+    assert dp(x*(y + 2*KD(i, j)), (j, 1, 3)) == (x*y)**3 + \
+        2*x*(x*y)**2*KD(i, 1) + 2*x*y*x*x*y*KD(i, 2) + 2*(x*y)**2*x*KD(i, 3)
+    assert dp(x*(y + 2*KD(i, j)), (j, 1, 1)) == x*(y + 2*KD(i, 1))
+    assert dp(x*(y + 2*KD(i, j)), (j, 2, 2)) == x*(y + 2*KD(i, 2))
+    assert dp(x*(y + 2*KD(i, j)), (j, 3, 3)) == x*(y + 2*KD(i, 3))
+    assert dp(x*(y + 2*KD(i, j)), (j, 1, k)) == \
+        (x*y)**k + Piecewise(
+            (2*(x*y)**(i - 1)*x*(x*y)**(k - i), And(1 <= i, i <= k)),
+            (0, True)
+        ).expand()
+    assert dp(x*(y + 2*KD(i, j)), (j, k, 3)) == \
+        ((x*y)**(-k + 4) + Piecewise(
+            (2*(x*y)**(i - k)*x*(x*y)**(3 - i), And(k <= i, i <= 3)),
+            (0, True)
+        )).expand()
+    assert dp(x*(y + 2*KD(i, j)), (j, k, l)) == \
+        ((x*y)**(-k + l + 1) + Piecewise(
+            (2*(x*y)**(i - k)*x*(x*y)**(l - i), And(k <= i, i <= l)),
+            (0, True)
+        )).expand()
+
+
+def test_deltaproduct_mul_add_x_y_add_y_kd():
+    assert dp((x + y)*(y + KD(i, j)), (j, 1, 3)) == ((x + y)*y)**3 + \
+        (x + y)*((x + y)*y)**2*KD(i, 1) + \
+        (x + y)*y*(x + y)**2*y*KD(i, 2) + \
+        ((x + y)*y)**2*(x + y)*KD(i, 3)
+    assert dp((x + y)*(y + KD(i, j)), (j, 1, 1)) == (x + y)*(y + KD(i, 1))
+    assert dp((x + y)*(y + KD(i, j)), (j, 2, 2)) == (x + y)*(y + KD(i, 2))
+    assert dp((x + y)*(y + KD(i, j)), (j, 3, 3)) == (x + y)*(y + KD(i, 3))
+    assert dp((x + y)*(y + KD(i, j)), (j, 1, k)) == \
+        ((x + y)*y)**k + Piecewise(
+            (((x + y)*y)**(-1)*((x + y)*y)**i*(x + y)*((x + y)*y
+            )**k*((x + y)*y)**(-i), (i >= 1) & (i <= k)), (0, True))
+    assert dp((x + y)*(y + KD(i, j)), (j, k, 3)) == (
+        (x + y)*y)**4*((x + y)*y)**(-k) + Piecewise((((x + y)*y)**i*(
+        (x + y)*y)**(-k)*(x + y)*((x + y)*y)**3*((x + y)*y)**(-i),
+        (i >= k) & (i <= 3)), (0, True))
+    assert dp((x + y)*(y + KD(i, j)), (j, k, l)) == \
+        (x + y)*y*((x + y)*y)**l*((x + y)*y)**(-k) + Piecewise(
+        (((x + y)*y)**i*((x + y)*y)**(-k)*(x + y)*((x + y)*y
+        )**l*((x + y)*y)**(-i), (i >= k) & (i <= l)), (0, True))
+
+
+def test_deltaproduct_mul_add_x_kd_add_y_kd():
+    assert dp((x + KD(i, k))*(y + KD(i, j)), (j, 1, 3)) == \
+        KD(i, 1)*(KD(i, k) + x)*((KD(i, k) + x)*y)**2 + \
+        KD(i, 2)*(KD(i, k) + x)*y*(KD(i, k) + x)**2*y + \
+        KD(i, 3)*((KD(i, k) + x)*y)**2*(KD(i, k) + x) + \
+        ((KD(i, k) + x)*y)**3
+    assert dp((x + KD(i, k))*(y + KD(i, j)), (j, 1, 1)) == \
+        (x + KD(i, k))*(y + KD(i, 1))
+    assert dp((x + KD(i, k))*(y + KD(i, j)), (j, 2, 2)) == \
+        (x + KD(i, k))*(y + KD(i, 2))
+    assert dp((x + KD(i, k))*(y + KD(i, j)), (j, 3, 3)) == \
+        (x + KD(i, k))*(y + KD(i, 3))
+    assert dp((x + KD(i, k))*(y + KD(i, j)), (j, 1, k)) == \
+        ((KD(i, k) + x)*y)**k + Piecewise(
+        (((KD(i, k) + x)*y)**(-1)*((KD(i, k) + x)*y)**i*(KD(i, k) + x
+        )*((KD(i, k) + x)*y)**k*((KD(i, k) + x)*y)**(-i), (i >= 1
+        ) & (i <= k)), (0, True))
+    assert dp((x + KD(i, k))*(y + KD(i, j)), (j, k, 3)) == (
+        (KD(i, k) + x)*y)**4*((KD(i, k) + x)*y)**(-k) + Piecewise(
+        (((KD(i, k) + x)*y)**i*((KD(i, k) + x)*y)**(-k)*(KD(i, k)
+        + x)*((KD(i, k) + x)*y)**3*((KD(i, k) + x)*y)**(-i),
+        (i >= k) & (i <= 3)), (0, True))
+    assert dp((x + KD(i, k))*(y + KD(i, j)), (j, k, l)) == (
+        KD(i, k) + x)*y*((KD(i, k) + x)*y)**l*((KD(i, k) + x)*y
+        )**(-k) + Piecewise((((KD(i, k) + x)*y)**i*((KD(i, k) + x
+        )*y)**(-k)*(KD(i, k) + x)*((KD(i, k) + x)*y)**l*((KD(i, k) + x
+        )*y)**(-i), (i >= k) & (i <= l)), (0, True))
+
+
+def test_deltasummation_trivial():
+    assert ds(x, (j, 1, 0)) == 0
+    assert ds(x, (j, 1, 3)) == 3*x
+    assert ds(x + y, (j, 1, 3)) == 3*(x + y)
+    assert ds(x*y, (j, 1, 3)) == 3*x*y
+    assert ds(KD(i, j), (k, 1, 3)) == 3*KD(i, j)
+    assert ds(x*KD(i, j), (k, 1, 3)) == 3*x*KD(i, j)
+    assert ds(x*y*KD(i, j), (k, 1, 3)) == 3*x*y*KD(i, j)
+
+
+def test_deltasummation_basic_numerical():
+    n = symbols('n', integer=True, nonzero=True)
+    assert ds(KD(n, 0), (n, 1, 3)) == 0
+
+    # return unevaluated, until it gets implemented
+    assert ds(KD(i**2, j**2), (j, -oo, oo)) == \
+        Sum(KD(i**2, j**2), (j, -oo, oo))
+
+    assert Piecewise((KD(i, k), And(1 <= i, i <= 3)), (0, True)) == \
+        ds(KD(i, j)*KD(j, k), (j, 1, 3)) == \
+        ds(KD(j, k)*KD(i, j), (j, 1, 3))
+
+    assert ds(KD(i, k), (k, -oo, oo)) == 1
+    assert ds(KD(i, k), (k, 0, oo)) == Piecewise((1, S.Zero <= i), (0, True))
+    assert ds(KD(i, k), (k, 1, 3)) == \
+        Piecewise((1, And(1 <= i, i <= 3)), (0, True))
+    assert ds(k*KD(i, j)*KD(j, k), (k, -oo, oo)) == j*KD(i, j)
+    assert ds(j*KD(i, j), (j, -oo, oo)) == i
+    assert ds(i*KD(i, j), (i, -oo, oo)) == j
+    assert ds(x, (i, 1, 3)) == 3*x
+    assert ds((i + j)*KD(i, j), (j, -oo, oo)) == 2*i
+
+
+def test_deltasummation_basic_symbolic():
+    assert ds(KD(i, j), (j, 1, 3)) == \
+        Piecewise((1, And(1 <= i, i <= 3)), (0, True))
+    assert ds(KD(i, j), (j, 1, 1)) == Piecewise((1, Eq(i, 1)), (0, True))
+    assert ds(KD(i, j), (j, 2, 2)) == Piecewise((1, Eq(i, 2)), (0, True))
+    assert ds(KD(i, j), (j, 3, 3)) == Piecewise((1, Eq(i, 3)), (0, True))
+    assert ds(KD(i, j), (j, 1, k)) == \
+        Piecewise((1, And(1 <= i, i <= k)), (0, True))
+    assert ds(KD(i, j), (j, k, 3)) == \
+        Piecewise((1, And(k <= i, i <= 3)), (0, True))
+    assert ds(KD(i, j), (j, k, l)) == \
+        Piecewise((1, And(k <= i, i <= l)), (0, True))
+
+
+def test_deltasummation_mul_x_kd():
+    assert ds(x*KD(i, j), (j, 1, 3)) == \
+        Piecewise((x, And(1 <= i, i <= 3)), (0, True))
+    assert ds(x*KD(i, j), (j, 1, 1)) == Piecewise((x, Eq(i, 1)), (0, True))
+    assert ds(x*KD(i, j), (j, 2, 2)) == Piecewise((x, Eq(i, 2)), (0, True))
+    assert ds(x*KD(i, j), (j, 3, 3)) == Piecewise((x, Eq(i, 3)), (0, True))
+    assert ds(x*KD(i, j), (j, 1, k)) == \
+        Piecewise((x, And(1 <= i, i <= k)), (0, True))
+    assert ds(x*KD(i, j), (j, k, 3)) == \
+        Piecewise((x, And(k <= i, i <= 3)), (0, True))
+    assert ds(x*KD(i, j), (j, k, l)) == \
+        Piecewise((x, And(k <= i, i <= l)), (0, True))
+
+
+def test_deltasummation_mul_add_x_y_kd():
+    assert ds((x + y)*KD(i, j), (j, 1, 3)) == \
+        Piecewise((x + y, And(1 <= i, i <= 3)), (0, True))
+    assert ds((x + y)*KD(i, j), (j, 1, 1)) == \
+        Piecewise((x + y, Eq(i, 1)), (0, True))
+    assert ds((x + y)*KD(i, j), (j, 2, 2)) == \
+        Piecewise((x + y, Eq(i, 2)), (0, True))
+    assert ds((x + y)*KD(i, j), (j, 3, 3)) == \
+        Piecewise((x + y, Eq(i, 3)), (0, True))
+    assert ds((x + y)*KD(i, j), (j, 1, k)) == \
+        Piecewise((x + y, And(1 <= i, i <= k)), (0, True))
+    assert ds((x + y)*KD(i, j), (j, k, 3)) == \
+        Piecewise((x + y, And(k <= i, i <= 3)), (0, True))
+    assert ds((x + y)*KD(i, j), (j, k, l)) == \
+        Piecewise((x + y, And(k <= i, i <= l)), (0, True))
+
+
+def test_deltasummation_add_kd_kd():
+    assert ds(KD(i, k) + KD(j, k), (k, 1, 3)) == piecewise_fold(
+        Piecewise((1, And(1 <= i, i <= 3)), (0, True)) +
+        Piecewise((1, And(1 <= j, j <= 3)), (0, True)))
+    assert ds(KD(i, k) + KD(j, k), (k, 1, 1)) == piecewise_fold(
+        Piecewise((1, Eq(i, 1)), (0, True)) +
+        Piecewise((1, Eq(j, 1)), (0, True)))
+    assert ds(KD(i, k) + KD(j, k), (k, 2, 2)) == piecewise_fold(
+        Piecewise((1, Eq(i, 2)), (0, True)) +
+        Piecewise((1, Eq(j, 2)), (0, True)))
+    assert ds(KD(i, k) + KD(j, k), (k, 3, 3)) == piecewise_fold(
+        Piecewise((1, Eq(i, 3)), (0, True)) +
+        Piecewise((1, Eq(j, 3)), (0, True)))
+    assert ds(KD(i, k) + KD(j, k), (k, 1, l)) == piecewise_fold(
+        Piecewise((1, And(1 <= i, i <= l)), (0, True)) +
+        Piecewise((1, And(1 <= j, j <= l)), (0, True)))
+    assert ds(KD(i, k) + KD(j, k), (k, l, 3)) == piecewise_fold(
+        Piecewise((1, And(l <= i, i <= 3)), (0, True)) +
+        Piecewise((1, And(l <= j, j <= 3)), (0, True)))
+    assert ds(KD(i, k) + KD(j, k), (k, l, m)) == piecewise_fold(
+        Piecewise((1, And(l <= i, i <= m)), (0, True)) +
+        Piecewise((1, And(l <= j, j <= m)), (0, True)))
+
+
+def test_deltasummation_add_mul_x_kd_kd():
+    assert ds(x*KD(i, k) + KD(j, k), (k, 1, 3)) == piecewise_fold(
+        Piecewise((x, And(1 <= i, i <= 3)), (0, True)) +
+        Piecewise((1, And(1 <= j, j <= 3)), (0, True)))
+    assert ds(x*KD(i, k) + KD(j, k), (k, 1, 1)) == piecewise_fold(
+        Piecewise((x, Eq(i, 1)), (0, True)) +
+        Piecewise((1, Eq(j, 1)), (0, True)))
+    assert ds(x*KD(i, k) + KD(j, k), (k, 2, 2)) == piecewise_fold(
+        Piecewise((x, Eq(i, 2)), (0, True)) +
+        Piecewise((1, Eq(j, 2)), (0, True)))
+    assert ds(x*KD(i, k) + KD(j, k), (k, 3, 3)) == piecewise_fold(
+        Piecewise((x, Eq(i, 3)), (0, True)) +
+        Piecewise((1, Eq(j, 3)), (0, True)))
+    assert ds(x*KD(i, k) + KD(j, k), (k, 1, l)) == piecewise_fold(
+        Piecewise((x, And(1 <= i, i <= l)), (0, True)) +
+        Piecewise((1, And(1 <= j, j <= l)), (0, True)))
+    assert ds(x*KD(i, k) + KD(j, k), (k, l, 3)) == piecewise_fold(
+        Piecewise((x, And(l <= i, i <= 3)), (0, True)) +
+        Piecewise((1, And(l <= j, j <= 3)), (0, True)))
+    assert ds(x*KD(i, k) + KD(j, k), (k, l, m)) == piecewise_fold(
+        Piecewise((x, And(l <= i, i <= m)), (0, True)) +
+        Piecewise((1, And(l <= j, j <= m)), (0, True)))
+
+
+def test_deltasummation_mul_x_add_kd_kd():
+    assert ds(x*(KD(i, k) + KD(j, k)), (k, 1, 3)) == piecewise_fold(
+        Piecewise((x, And(1 <= i, i <= 3)), (0, True)) +
+        Piecewise((x, And(1 <= j, j <= 3)), (0, True)))
+    assert ds(x*(KD(i, k) + KD(j, k)), (k, 1, 1)) == piecewise_fold(
+        Piecewise((x, Eq(i, 1)), (0, True)) +
+        Piecewise((x, Eq(j, 1)), (0, True)))
+    assert ds(x*(KD(i, k) + KD(j, k)), (k, 2, 2)) == piecewise_fold(
+        Piecewise((x, Eq(i, 2)), (0, True)) +
+        Piecewise((x, Eq(j, 2)), (0, True)))
+    assert ds(x*(KD(i, k) + KD(j, k)), (k, 3, 3)) == piecewise_fold(
+        Piecewise((x, Eq(i, 3)), (0, True)) +
+        Piecewise((x, Eq(j, 3)), (0, True)))
+    assert ds(x*(KD(i, k) + KD(j, k)), (k, 1, l)) == piecewise_fold(
+        Piecewise((x, And(1 <= i, i <= l)), (0, True)) +
+        Piecewise((x, And(1 <= j, j <= l)), (0, True)))
+    assert ds(x*(KD(i, k) + KD(j, k)), (k, l, 3)) == piecewise_fold(
+        Piecewise((x, And(l <= i, i <= 3)), (0, True)) +
+        Piecewise((x, And(l <= j, j <= 3)), (0, True)))
+    assert ds(x*(KD(i, k) + KD(j, k)), (k, l, m)) == piecewise_fold(
+        Piecewise((x, And(l <= i, i <= m)), (0, True)) +
+        Piecewise((x, And(l <= j, j <= m)), (0, True)))
+
+
+def test_deltasummation_mul_add_x_y_add_kd_kd():
+    assert ds((x + y)*(KD(i, k) + KD(j, k)), (k, 1, 3)) == piecewise_fold(
+        Piecewise((x + y, And(1 <= i, i <= 3)), (0, True)) +
+        Piecewise((x + y, And(1 <= j, j <= 3)), (0, True)))
+    assert ds((x + y)*(KD(i, k) + KD(j, k)), (k, 1, 1)) == piecewise_fold(
+        Piecewise((x + y, Eq(i, 1)), (0, True)) +
+        Piecewise((x + y, Eq(j, 1)), (0, True)))
+    assert ds((x + y)*(KD(i, k) + KD(j, k)), (k, 2, 2)) == piecewise_fold(
+        Piecewise((x + y, Eq(i, 2)), (0, True)) +
+        Piecewise((x + y, Eq(j, 2)), (0, True)))
+    assert ds((x + y)*(KD(i, k) + KD(j, k)), (k, 3, 3)) == piecewise_fold(
+        Piecewise((x + y, Eq(i, 3)), (0, True)) +
+        Piecewise((x + y, Eq(j, 3)), (0, True)))
+    assert ds((x + y)*(KD(i, k) + KD(j, k)), (k, 1, l)) == piecewise_fold(
+        Piecewise((x + y, And(1 <= i, i <= l)), (0, True)) +
+        Piecewise((x + y, And(1 <= j, j <= l)), (0, True)))
+    assert ds((x + y)*(KD(i, k) + KD(j, k)), (k, l, 3)) == piecewise_fold(
+        Piecewise((x + y, And(l <= i, i <= 3)), (0, True)) +
+        Piecewise((x + y, And(l <= j, j <= 3)), (0, True)))
+    assert ds((x + y)*(KD(i, k) + KD(j, k)), (k, l, m)) == piecewise_fold(
+        Piecewise((x + y, And(l <= i, i <= m)), (0, True)) +
+        Piecewise((x + y, And(l <= j, j <= m)), (0, True)))
+
+
+def test_deltasummation_add_mul_x_y_mul_x_kd():
+    assert ds(x*y + x*KD(i, j), (j, 1, 3)) == \
+        Piecewise((3*x*y + x, And(1 <= i, i <= 3)), (3*x*y, True))
+    assert ds(x*y + x*KD(i, j), (j, 1, 1)) == \
+        Piecewise((x*y + x, Eq(i, 1)), (x*y, True))
+    assert ds(x*y + x*KD(i, j), (j, 2, 2)) == \
+        Piecewise((x*y + x, Eq(i, 2)), (x*y, True))
+    assert ds(x*y + x*KD(i, j), (j, 3, 3)) == \
+        Piecewise((x*y + x, Eq(i, 3)), (x*y, True))
+    assert ds(x*y + x*KD(i, j), (j, 1, k)) == \
+        Piecewise((k*x*y + x, And(1 <= i, i <= k)), (k*x*y, True))
+    assert ds(x*y + x*KD(i, j), (j, k, 3)) == \
+        Piecewise(((4 - k)*x*y + x, And(k <= i, i <= 3)), ((4 - k)*x*y, True))
+    assert ds(x*y + x*KD(i, j), (j, k, l)) == Piecewise(
+        ((l - k + 1)*x*y + x, And(k <= i, i <= l)), ((l - k + 1)*x*y, True))
+
+
+def test_deltasummation_mul_x_add_y_kd():
+    assert ds(x*(y + KD(i, j)), (j, 1, 3)) == \
+        Piecewise((3*x*y + x, And(1 <= i, i <= 3)), (3*x*y, True))
+    assert ds(x*(y + KD(i, j)), (j, 1, 1)) == \
+        Piecewise((x*y + x, Eq(i, 1)), (x*y, True))
+    assert ds(x*(y + KD(i, j)), (j, 2, 2)) == \
+        Piecewise((x*y + x, Eq(i, 2)), (x*y, True))
+    assert ds(x*(y + KD(i, j)), (j, 3, 3)) == \
+        Piecewise((x*y + x, Eq(i, 3)), (x*y, True))
+    assert ds(x*(y + KD(i, j)), (j, 1, k)) == \
+        Piecewise((k*x*y + x, And(1 <= i, i <= k)), (k*x*y, True))
+    assert ds(x*(y + KD(i, j)), (j, k, 3)) == \
+        Piecewise(((4 - k)*x*y + x, And(k <= i, i <= 3)), ((4 - k)*x*y, True))
+    assert ds(x*(y + KD(i, j)), (j, k, l)) == Piecewise(
+        ((l - k + 1)*x*y + x, And(k <= i, i <= l)), ((l - k + 1)*x*y, True))
+
+
+def test_deltasummation_mul_x_add_y_twokd():
+    assert ds(x*(y + 2*KD(i, j)), (j, 1, 3)) == \
+        Piecewise((3*x*y + 2*x, And(1 <= i, i <= 3)), (3*x*y, True))
+    assert ds(x*(y + 2*KD(i, j)), (j, 1, 1)) == \
+        Piecewise((x*y + 2*x, Eq(i, 1)), (x*y, True))
+    assert ds(x*(y + 2*KD(i, j)), (j, 2, 2)) == \
+        Piecewise((x*y + 2*x, Eq(i, 2)), (x*y, True))
+    assert ds(x*(y + 2*KD(i, j)), (j, 3, 3)) == \
+        Piecewise((x*y + 2*x, Eq(i, 3)), (x*y, True))
+    assert ds(x*(y + 2*KD(i, j)), (j, 1, k)) == \
+        Piecewise((k*x*y + 2*x, And(1 <= i, i <= k)), (k*x*y, True))
+    assert ds(x*(y + 2*KD(i, j)), (j, k, 3)) == Piecewise(
+        ((4 - k)*x*y + 2*x, And(k <= i, i <= 3)), ((4 - k)*x*y, True))
+    assert ds(x*(y + 2*KD(i, j)), (j, k, l)) == Piecewise(
+        ((l - k + 1)*x*y + 2*x, And(k <= i, i <= l)), ((l - k + 1)*x*y, True))
+
+
+def test_deltasummation_mul_add_x_y_add_y_kd():
+    assert ds((x + y)*(y + KD(i, j)), (j, 1, 3)) == Piecewise(
+        (3*(x + y)*y + x + y, And(1 <= i, i <= 3)), (3*(x + y)*y, True))
+    assert ds((x + y)*(y + KD(i, j)), (j, 1, 1)) == \
+        Piecewise(((x + y)*y + x + y, Eq(i, 1)), ((x + y)*y, True))
+    assert ds((x + y)*(y + KD(i, j)), (j, 2, 2)) == \
+        Piecewise(((x + y)*y + x + y, Eq(i, 2)), ((x + y)*y, True))
+    assert ds((x + y)*(y + KD(i, j)), (j, 3, 3)) == \
+        Piecewise(((x + y)*y + x + y, Eq(i, 3)), ((x + y)*y, True))
+    assert ds((x + y)*(y + KD(i, j)), (j, 1, k)) == Piecewise(
+        (k*(x + y)*y + x + y, And(1 <= i, i <= k)), (k*(x + y)*y, True))
+    assert ds((x + y)*(y + KD(i, j)), (j, k, 3)) == Piecewise(
+        ((4 - k)*(x + y)*y + x + y, And(k <= i, i <= 3)),
+        ((4 - k)*(x + y)*y, True))
+    assert ds((x + y)*(y + KD(i, j)), (j, k, l)) == Piecewise(
+        ((l - k + 1)*(x + y)*y + x + y, And(k <= i, i <= l)),
+        ((l - k + 1)*(x + y)*y, True))
+
+
+def test_deltasummation_mul_add_x_kd_add_y_kd():
+    assert ds((x + KD(i, k))*(y + KD(i, j)), (j, 1, 3)) == piecewise_fold(
+        Piecewise((KD(i, k) + x, And(1 <= i, i <= 3)), (0, True)) +
+        3*(KD(i, k) + x)*y)
+    assert ds((x + KD(i, k))*(y + KD(i, j)), (j, 1, 1)) == piecewise_fold(
+        Piecewise((KD(i, k) + x, Eq(i, 1)), (0, True)) +
+        (KD(i, k) + x)*y)
+    assert ds((x + KD(i, k))*(y + KD(i, j)), (j, 2, 2)) == piecewise_fold(
+        Piecewise((KD(i, k) + x, Eq(i, 2)), (0, True)) +
+        (KD(i, k) + x)*y)
+    assert ds((x + KD(i, k))*(y + KD(i, j)), (j, 3, 3)) == piecewise_fold(
+        Piecewise((KD(i, k) + x, Eq(i, 3)), (0, True)) +
+        (KD(i, k) + x)*y)
+    assert ds((x + KD(i, k))*(y + KD(i, j)), (j, 1, k)) == piecewise_fold(
+        Piecewise((KD(i, k) + x, And(1 <= i, i <= k)), (0, True)) +
+        k*(KD(i, k) + x)*y)
+    assert ds((x + KD(i, k))*(y + KD(i, j)), (j, k, 3)) == piecewise_fold(
+        Piecewise((KD(i, k) + x, And(k <= i, i <= 3)), (0, True)) +
+        (4 - k)*(KD(i, k) + x)*y)
+    assert ds((x + KD(i, k))*(y + KD(i, j)), (j, k, l)) == piecewise_fold(
+        Piecewise((KD(i, k) + x, And(k <= i, i <= l)), (0, True)) +
+        (l - k + 1)*(KD(i, k) + x)*y)
+
+
+def test_extract_delta():
+    raises(ValueError, lambda: _extract_delta(KD(i, j) + KD(k, l), i))

.venv/lib/python3.13/site-packages/sympy/concrete/tests/test_gosper.py

@@ -0,0 +1,204 @@
+"""Tests for Gosper's algorithm for hypergeometric summation. """
+
+from sympy.core.numbers import (Rational, pi)
+from sympy.core.singleton import S
+from sympy.core.symbol import Symbol
+from sympy.functions.combinatorial.factorials import (binomial, factorial)
+from sympy.functions.elementary.exponential import (exp, log)
+from sympy.functions.elementary.miscellaneous import sqrt
+from sympy.functions.special.gamma_functions import gamma
+from sympy.polys.polytools import Poly
+from sympy.simplify.simplify import simplify
+from sympy.concrete.gosper import gosper_normal, gosper_sum, gosper_term
+from sympy.abc import a, b, j, k, m, n, r, x
+
+
+def test_gosper_normal():
+    eq = 4*n + 5, 2*(4*n + 1)*(2*n + 3), n
+    assert gosper_normal(*eq) == \
+        (Poly(Rational(1, 4), n), Poly(n + Rational(3, 2)), Poly(n + Rational(1, 4)))
+    assert gosper_normal(*eq, polys=False) == \
+        (Rational(1, 4), n + Rational(3, 2), n + Rational(1, 4))
+
+
+def test_gosper_term():
+    assert gosper_term((4*k + 1)*factorial(
+        k)/factorial(2*k + 1), k) == (-k - S.Half)/(k + Rational(1, 4))
+
+
+def test_gosper_sum():
+    assert gosper_sum(1, (k, 0, n)) == 1 + n
+    assert gosper_sum(k, (k, 0, n)) == n*(1 + n)/2
+    assert gosper_sum(k**2, (k, 0, n)) == n*(1 + n)*(1 + 2*n)/6
+    assert gosper_sum(k**3, (k, 0, n)) == n**2*(1 + n)**2/4
+
+    assert gosper_sum(2**k, (k, 0, n)) == 2*2**n - 1
+
+    assert gosper_sum(factorial(k), (k, 0, n)) is None
+    assert gosper_sum(binomial(n, k), (k, 0, n)) is None
+
+    assert gosper_sum(factorial(k)/k**2, (k, 0, n)) is None
+    assert gosper_sum((k - 3)*factorial(k), (k, 0, n)) is None
+
+    assert gosper_sum(k*factorial(k), k) == factorial(k)
+    assert gosper_sum(
+        k*factorial(k), (k, 0, n)) == n*factorial(n) + factorial(n) - 1
+
+    assert gosper_sum((-1)**k*binomial(n, k), (k, 0, n)) == 0
+    assert gosper_sum((
+        -1)**k*binomial(n, k), (k, 0, m)) == -(-1)**m*(m - n)*binomial(n, m)/n
+
+    assert gosper_sum((4*k + 1)*factorial(k)/factorial(2*k + 1), (k, 0, n)) == \
+        (2*factorial(2*n + 1) - factorial(n))/factorial(2*n + 1)
+
+    # issue 6033:
+    assert gosper_sum(
+        n*(n + a + b)*a**n*b**n/(factorial(n + a)*factorial(n + b)), \
+        (n, 0, m)).simplify() == -exp(m*log(a) + m*log(b))*gamma(a + 1) \
+        *gamma(b + 1)/(gamma(a)*gamma(b)*gamma(a + m + 1)*gamma(b + m + 1)) \
+        + 1/(gamma(a)*gamma(b))
+
+
+def test_gosper_sum_indefinite():
+    assert gosper_sum(k, k) == k*(k - 1)/2
+    assert gosper_sum(k**2, k) == k*(k - 1)*(2*k - 1)/6
+
+    assert gosper_sum(1/(k*(k + 1)), k) == -1/k
+    assert gosper_sum(-(27*k**4 + 158*k**3 + 430*k**2 + 678*k + 445)*gamma(2*k
+                      + 4)/(3*(3*k + 7)*gamma(3*k + 6)), k) == \
+        (3*k + 5)*(k**2 + 2*k + 5)*gamma(2*k + 4)/gamma(3*k + 6)
+
+
+def test_gosper_sum_parametric():
+    assert gosper_sum(binomial(S.Half, m - j + 1)*binomial(S.Half, m + j), (j, 1, n)) == \
+        n*(1 + m - n)*(-1 + 2*m + 2*n)*binomial(S.Half, 1 + m - n)* \
+        binomial(S.Half, m + n)/(m*(1 + 2*m))
+
+
+def test_gosper_sum_algebraic():
+    assert gosper_sum(
+        n**2 + sqrt(2), (n, 0, m)) == (m + 1)*(2*m**2 + m + 6*sqrt(2))/6
+
+
+def test_gosper_sum_iterated():
+    f1 = binomial(2*k, k)/4**k
+    f2 = (1 + 2*n)*binomial(2*n, n)/4**n
+    f3 = (1 + 2*n)*(3 + 2*n)*binomial(2*n, n)/(3*4**n)
+    f4 = (1 + 2*n)*(3 + 2*n)*(5 + 2*n)*binomial(2*n, n)/(15*4**n)
+    f5 = (1 + 2*n)*(3 + 2*n)*(5 + 2*n)*(7 + 2*n)*binomial(2*n, n)/(105*4**n)
+
+    assert gosper_sum(f1, (k, 0, n)) == f2
+    assert gosper_sum(f2, (n, 0, n)) == f3
+    assert gosper_sum(f3, (n, 0, n)) == f4
+    assert gosper_sum(f4, (n, 0, n)) == f5
+
+# the AeqB tests test expressions given in
+# www.math.upenn.edu/~wilf/AeqB.pdf
+
+
+def test_gosper_sum_AeqB_part1():
+    f1a = n**4
+    f1b = n**3*2**n
+    f1c = 1/(n**2 + sqrt(5)*n - 1)
+    f1d = n**4*4**n/binomial(2*n, n)
+    f1e = factorial(3*n)/(factorial(n)*factorial(n + 1)*factorial(n + 2)*27**n)
+    f1f = binomial(2*n, n)**2/((n + 1)*4**(2*n))
+    f1g = (4*n - 1)*binomial(2*n, n)**2/((2*n - 1)**2*4**(2*n))
+    f1h = n*factorial(n - S.Half)**2/factorial(n + 1)**2
+
+    g1a = m*(m + 1)*(2*m + 1)*(3*m**2 + 3*m - 1)/30
+    g1b = 26 + 2**(m + 1)*(m**3 - 3*m**2 + 9*m - 13)
+    g1c = (m + 1)*(m*(m**2 - 7*m + 3)*sqrt(5) - (
+        3*m**3 - 7*m**2 + 19*m - 6))/(2*m**3*sqrt(5) + m**4 + 5*m**2 - 1)/6
+    g1d = Rational(-2, 231) + 2*4**m*(m + 1)*(63*m**4 + 112*m**3 + 18*m**2 -
+             22*m + 3)/(693*binomial(2*m, m))
+    g1e = Rational(-9, 2) + (81*m**2 + 261*m + 200)*factorial(
+        3*m + 2)/(40*27**m*factorial(m)*factorial(m + 1)*factorial(m + 2))
+    g1f = (2*m + 1)**2*binomial(2*m, m)**2/(4**(2*m)*(m + 1))
+    g1g = -binomial(2*m, m)**2/4**(2*m)
+    g1h = 4*pi -(2*m + 1)**2*(3*m + 4)*factorial(m - S.Half)**2/factorial(m + 1)**2
+
+    g = gosper_sum(f1a, (n, 0, m))
+    assert g is not None and simplify(g - g1a) == 0
+    g = gosper_sum(f1b, (n, 0, m))
+    assert g is not None and simplify(g - g1b) == 0
+    g = gosper_sum(f1c, (n, 0, m))
+    assert g is not None and simplify(g - g1c) == 0
+    g = gosper_sum(f1d, (n, 0, m))
+    assert g is not None and simplify(g - g1d) == 0
+    g = gosper_sum(f1e, (n, 0, m))
+    assert g is not None and simplify(g - g1e) == 0
+    g = gosper_sum(f1f, (n, 0, m))
+    assert g is not None and simplify(g - g1f) == 0
+    g = gosper_sum(f1g, (n, 0, m))
+    assert g is not None and simplify(g - g1g) == 0
+    g = gosper_sum(f1h, (n, 0, m))
+    # need to call rewrite(gamma) here because we have terms involving
+    # factorial(1/2)
+    assert g is not None and simplify(g - g1h).rewrite(gamma) == 0
+
+
+def test_gosper_sum_AeqB_part2():
+    f2a = n**2*a**n
+    f2b = (n - r/2)*binomial(r, n)
+    f2c = factorial(n - 1)**2/(factorial(n - x)*factorial(n + x))
+
+    g2a = -a*(a + 1)/(a - 1)**3 + a**(
+        m + 1)*(a**2*m**2 - 2*a*m**2 + m**2 - 2*a*m + 2*m + a + 1)/(a - 1)**3
+    g2b = (m - r)*binomial(r, m)/2
+    ff = factorial(1 - x)*factorial(1 + x)
+    g2c = 1/ff*(
+        1 - 1/x**2) + factorial(m)**2/(x**2*factorial(m - x)*factorial(m + x))
+
+    g = gosper_sum(f2a, (n, 0, m))
+    assert g is not None and simplify(g - g2a) == 0
+    g = gosper_sum(f2b, (n, 0, m))
+    assert g is not None and simplify(g - g2b) == 0
+    g = gosper_sum(f2c, (n, 1, m))
+    assert g is not None and simplify(g - g2c) == 0
+
+
+def test_gosper_nan():
+    a = Symbol('a', positive=True)
+    b = Symbol('b', positive=True)
+    n = Symbol('n', integer=True)
+    m = Symbol('m', integer=True)
+    f2d = n*(n + a + b)*a**n*b**n/(factorial(n + a)*factorial(n + b))
+    g2d = 1/(factorial(a - 1)*factorial(
+        b - 1)) - a**(m + 1)*b**(m + 1)/(factorial(a + m)*factorial(b + m))
+    g = gosper_sum(f2d, (n, 0, m))
+    assert simplify(g - g2d) == 0
+
+
+def test_gosper_sum_AeqB_part3():
+    f3a = 1/n**4
+    f3b = (6*n + 3)/(4*n**4 + 8*n**3 + 8*n**2 + 4*n + 3)
+    f3c = 2**n*(n**2 - 2*n - 1)/(n**2*(n + 1)**2)
+    f3d = n**2*4**n/((n + 1)*(n + 2))
+    f3e = 2**n/(n + 1)
+    f3f = 4*(n - 1)*(n**2 - 2*n - 1)/(n**2*(n + 1)**2*(n - 2)**2*(n - 3)**2)
+    f3g = (n**4 - 14*n**2 - 24*n - 9)*2**n/(n**2*(n + 1)**2*(n + 2)**2*
+           (n + 3)**2)
+
+    # g3a -> no closed form
+    g3b = m*(m + 2)/(2*m**2 + 4*m + 3)
+    g3c = 2**m/m**2 - 2
+    g3d = Rational(2, 3) + 4**(m + 1)*(m - 1)/(m + 2)/3
+    # g3e -> no closed form
+    g3f = -(Rational(-1, 16) + 1/((m - 2)**2*(m + 1)**2))  # the AeqB key is wrong
+    g3g = Rational(-2, 9) + 2**(m + 1)/((m + 1)**2*(m + 3)**2)
+
+    g = gosper_sum(f3a, (n, 1, m))
+    assert g is None
+    g = gosper_sum(f3b, (n, 1, m))
+    assert g is not None and simplify(g - g3b) == 0
+    g = gosper_sum(f3c, (n, 1, m - 1))
+    assert g is not None and simplify(g - g3c) == 0
+    g = gosper_sum(f3d, (n, 1, m))
+    assert g is not None and simplify(g - g3d) == 0
+    g = gosper_sum(f3e, (n, 0, m - 1))
+    assert g is None
+    g = gosper_sum(f3f, (n, 4, m))
+    assert g is not None and simplify(g - g3f) == 0
+    g = gosper_sum(f3g, (n, 1, m))
+    assert g is not None and simplify(g - g3g) == 0

.venv/lib/python3.13/site-packages/sympy/concrete/tests/test_guess.py

@@ -0,0 +1,82 @@
+from sympy.concrete.guess import (
+            find_simple_recurrence_vector,
+            find_simple_recurrence,
+            rationalize,
+            guess_generating_function_rational,
+            guess_generating_function,
+            guess
+        )
+from sympy.concrete.products import Product
+from sympy.core.function import Function
+from sympy.core.numbers import Rational
+from sympy.core.singleton import S
+from sympy.core.symbol import (Symbol, symbols)
+from sympy.core.sympify import sympify
+from sympy.functions.combinatorial.factorials import (RisingFactorial, factorial)
+from sympy.functions.combinatorial.numbers import fibonacci
+from sympy.functions.elementary.exponential import exp
+
+
+def test_find_simple_recurrence_vector():
+    assert find_simple_recurrence_vector(
+            [fibonacci(k) for k in range(12)]) == [1, -1, -1]
+
+
+def test_find_simple_recurrence():
+    a = Function('a')
+    n = Symbol('n')
+    assert find_simple_recurrence([fibonacci(k) for k in range(12)]) == (
+        -a(n) - a(n + 1) + a(n + 2))
+
+    f = Function('a')
+    i = Symbol('n')
+    a = [1, 1, 1]
+    for k in range(15): a.append(5*a[-1]-3*a[-2]+8*a[-3])
+    assert find_simple_recurrence(a, A=f, N=i) == (
+        -8*f(i) + 3*f(i + 1) - 5*f(i + 2) + f(i + 3))
+    assert find_simple_recurrence([0, 2, 15, 74, 12, 3, 0,
+                                    1, 2, 85, 4, 5, 63]) == 0
+
+
+def test_rationalize():
+    from mpmath import cos, pi, mpf
+    assert rationalize(cos(pi/3)) == S.Half
+    assert rationalize(mpf("0.333333333333333")) == Rational(1, 3)
+    assert rationalize(mpf("-0.333333333333333")) == Rational(-1, 3)
+    assert rationalize(pi, maxcoeff = 250) == Rational(355, 113)
+
+
+def test_guess_generating_function_rational():
+    x = Symbol('x')
+    assert guess_generating_function_rational([fibonacci(k)
+        for k in range(5, 15)]) == ((3*x + 5)/(-x**2 - x + 1))
+
+
+def test_guess_generating_function():
+    x = Symbol('x')
+    assert guess_generating_function([fibonacci(k)
+        for k in range(5, 15)])['ogf'] == ((3*x + 5)/(-x**2 - x + 1))
+    assert guess_generating_function(
+        [1, 2, 5, 14, 41, 124, 383, 1200, 3799, 12122, 38919])['ogf'] == (
+        (1/(x**4 + 2*x**2 - 4*x + 1))**S.Half)
+    assert guess_generating_function(sympify(
+       "[3/2, 11/2, 0, -121/2, -363/2, 121, 4719/2, 11495/2, -8712, -178717/2]")
+       )['ogf'] == (x + Rational(3, 2))/(11*x**2 - 3*x + 1)
+    assert guess_generating_function([factorial(k) for k in range(12)],
+       types=['egf'])['egf'] == 1/(-x + 1)
+    assert guess_generating_function([k+1 for k in range(12)],
+       types=['egf']) == {'egf': (x + 1)*exp(x), 'lgdegf': (x + 2)/(x + 1)}
+
+
+def test_guess():
+    i0, i1 = symbols('i0 i1')
+    assert guess([1, 2, 6, 24, 120], evaluate=False) == [Product(i1 + 1, (i1, 1, i0 - 1))]
+    assert guess([1, 2, 6, 24, 120]) == [RisingFactorial(2, i0 - 1)]
+    assert guess([1, 2, 7, 42, 429, 7436, 218348, 10850216], niter=4) == [
+        2**(i0 - 1)*(Rational(27, 16))**(i0**2/2 - 3*i0/2 +
+        1)*Product(RisingFactorial(Rational(5, 3), i1 - 1)*RisingFactorial(Rational(7, 3), i1
+        - 1)/(RisingFactorial(Rational(3, 2), i1 - 1)*RisingFactorial(Rational(5, 2), i1 -
+        1)), (i1, 1, i0 - 1))]
+    assert guess([1, 0, 2]) == []
+    x, y = symbols('x y')
+    assert guess([1, 2, 6, 24, 120], variables=[x, y]) == [RisingFactorial(2, x - 1)]

.venv/lib/python3.13/site-packages/sympy/concrete/tests/test_products.py

@@ -0,0 +1,410 @@
+from sympy.concrete.products import (Product, product)
+from sympy.concrete.summations import Sum
+from sympy.core.function import (Derivative, Function, diff)
+from sympy.core.numbers import (Rational, oo, pi)
+from sympy.core.singleton import S
+from sympy.core.symbol import (Dummy, Symbol, symbols)
+from sympy.functions.combinatorial.factorials import (rf, factorial)
+from sympy.functions.elementary.exponential import (exp, log)
+from sympy.functions.elementary.miscellaneous import sqrt
+from sympy.functions.elementary.trigonometric import (cos, sin)
+from sympy.functions.special.tensor_functions import KroneckerDelta
+from sympy.simplify.combsimp import combsimp
+from sympy.simplify.simplify import simplify
+from sympy.testing.pytest import raises
+
+a, k, n, m, x = symbols('a,k,n,m,x', integer=True)
+f = Function('f')
+
+
+def test_karr_convention():
+    # Test the Karr product convention that we want to hold.
+    # See his paper "Summation in Finite Terms" for a detailed
+    # reasoning why we really want exactly this definition.
+    # The convention is described for sums on page 309 and
+    # essentially in section 1.4, definition 3. For products
+    # we can find in analogy:
+    #
+    # \prod_{m <= i < n} f(i) 'has the obvious meaning'      for m < n
+    # \prod_{m <= i < n} f(i) = 0                            for m = n
+    # \prod_{m <= i < n} f(i) = 1 / \prod_{n <= i < m} f(i)  for m > n
+    #
+    # It is important to note that he defines all products with
+    # the upper limit being *exclusive*.
+    # In contrast, SymPy and the usual mathematical notation has:
+    #
+    # prod_{i = a}^b f(i) = f(a) * f(a+1) * ... * f(b-1) * f(b)
+    #
+    # with the upper limit *inclusive*. So translating between
+    # the two we find that:
+    #
+    # \prod_{m <= i < n} f(i) = \prod_{i = m}^{n-1} f(i)
+    #
+    # where we intentionally used two different ways to typeset the
+    # products and its limits.
+
+    i = Symbol("i", integer=True)
+    k = Symbol("k", integer=True)
+    j = Symbol("j", integer=True, positive=True)
+
+    # A simple example with a concrete factors and symbolic limits.
+
+    # The normal product: m = k and n = k + j and therefore m < n:
+    m = k
+    n = k + j
+
+    a = m
+    b = n - 1
+    S1 = Product(i**2, (i, a, b)).doit()
+
+    # The reversed product: m = k + j and n = k and therefore m > n:
+    m = k + j
+    n = k
+
+    a = m
+    b = n - 1
+    S2 = Product(i**2, (i, a, b)).doit()
+
+    assert S1 * S2 == 1
+
+    # Test the empty product: m = k and n = k and therefore m = n:
+    m = k
+    n = k
+
+    a = m
+    b = n - 1
+    Sz = Product(i**2, (i, a, b)).doit()
+
+    assert Sz == 1
+
+    # Another example this time with an unspecified factor and
+    # numeric limits. (We can not do both tests in the same example.)
+    f = Function("f")
+
+    # The normal product with m < n:
+    m = 2
+    n = 11
+
+    a = m
+    b = n - 1
+    S1 = Product(f(i), (i, a, b)).doit()
+
+    # The reversed product with m > n:
+    m = 11
+    n = 2
+
+    a = m
+    b = n - 1
+    S2 = Product(f(i), (i, a, b)).doit()
+
+    assert simplify(S1 * S2) == 1
+
+    # Test the empty product with m = n:
+    m = 5
+    n = 5
+
+    a = m
+    b = n - 1
+    Sz = Product(f(i), (i, a, b)).doit()
+
+    assert Sz == 1
+
+
+def test_karr_proposition_2a():
+    # Test Karr, page 309, proposition 2, part a
+    i, u, v = symbols('i u v', integer=True)
+
+    def test_the_product(m, n):
+        # g
+        g = i**3 + 2*i**2 - 3*i
+        # f = Delta g
+        f = simplify(g.subs(i, i+1) / g)
+        # The product
+        a = m
+        b = n - 1
+        P = Product(f, (i, a, b)).doit()
+        # Test if Product_{m <= i < n} f(i) = g(n) / g(m)
+        assert combsimp(P / (g.subs(i, n) / g.subs(i, m))) == 1
+
+    # m < n
+    test_the_product(u, u + v)
+    # m = n
+    test_the_product(u, u)
+    # m > n
+    test_the_product(u + v, u)
+
+
+def test_karr_proposition_2b():
+    # Test Karr, page 309, proposition 2, part b
+    i, u, v, w = symbols('i u v w', integer=True)
+
+    def test_the_product(l, n, m):
+        # Productmand
+        s = i**3
+        # First product
+        a = l
+        b = n - 1
+        S1 = Product(s, (i, a, b)).doit()
+        # Second product
+        a = l
+        b = m - 1
+        S2 = Product(s, (i, a, b)).doit()
+        # Third product
+        a = m
+        b = n - 1
+        S3 = Product(s, (i, a, b)).doit()
+        # Test if S1 = S2 * S3 as required
+        assert combsimp(S1 / (S2 * S3)) == 1
+
+    # l < m < n
+    test_the_product(u, u + v, u + v + w)
+    # l < m = n
+    test_the_product(u, u + v, u + v)
+    # l < m > n
+    test_the_product(u, u + v + w, v)
+    # l = m < n
+    test_the_product(u, u, u + v)
+    # l = m = n
+    test_the_product(u, u, u)
+    # l = m > n
+    test_the_product(u + v, u + v, u)
+    # l > m < n
+    test_the_product(u + v, u, u + w)
+    # l > m = n
+    test_the_product(u + v, u, u)
+    # l > m > n
+    test_the_product(u + v + w, u + v, u)
+
+
+def test_simple_products():
+    assert product(2, (k, a, n)) == 2**(n - a + 1)
+    assert product(k, (k, 1, n)) == factorial(n)
+    assert product(k**3, (k, 1, n)) == factorial(n)**3
+
+    assert product(k + 1, (k, 0, n - 1)) == factorial(n)
+    assert product(k + 1, (k, a, n - 1)) == rf(1 + a, n - a)
+
+    assert product(cos(k), (k, 0, 5)) == cos(1)*cos(2)*cos(3)*cos(4)*cos(5)
+    assert product(cos(k), (k, 3, 5)) == cos(3)*cos(4)*cos(5)
+    assert product(cos(k), (k, 1, Rational(5, 2))) != cos(1)*cos(2)
+
+    assert isinstance(product(k**k, (k, 1, n)), Product)
+
+    assert Product(x**k, (k, 1, n)).variables == [k]
+
+    raises(ValueError, lambda: Product(n))
+    raises(ValueError, lambda: Product(n, k))
+    raises(ValueError, lambda: Product(n, k, 1))
+    raises(ValueError, lambda: Product(n, k, 1, 10))
+    raises(ValueError, lambda: Product(n, (k, 1)))
+
+    assert product(1, (n, 1, oo)) == 1  # issue 8301
+    assert product(2, (n, 1, oo)) is oo
+    assert product(-1, (n, 1, oo)).func is Product
+
+
+def test_multiple_products():
+    assert product(x, (n, 1, k), (k, 1, m)) == x**(m**2/2 + m/2)
+    assert product(f(n), (
+        n, 1, m), (m, 1, k)) == Product(f(n), (n, 1, m), (m, 1, k)).doit()
+    assert Product(f(n), (m, 1, k), (n, 1, k)).doit() == \
+        Product(Product(f(n), (m, 1, k)), (n, 1, k)).doit() == \
+        product(f(n), (m, 1, k), (n, 1, k)) == \
+        product(product(f(n), (m, 1, k)), (n, 1, k)) == \
+        Product(f(n)**k, (n, 1, k))
+    assert Product(
+        x, (x, 1, k), (k, 1, n)).doit() == Product(factorial(k), (k, 1, n))
+
+    assert Product(x**k, (n, 1, k), (k, 1, m)).variables == [n, k]
+
+
+def test_rational_products():
+    assert product(1 + 1/k, (k, 1, n)) == rf(2, n)/factorial(n)
+
+
+def test_special_products():
+    # Wallis product
+    assert product((4*k)**2 / (4*k**2 - 1), (k, 1, n)) == \
+        4**n*factorial(n)**2/rf(S.Half, n)/rf(Rational(3, 2), n)
+
+    # Euler's product formula for sin
+    assert product(1 + a/k**2, (k, 1, n)) == \
+        rf(1 - sqrt(-a), n)*rf(1 + sqrt(-a), n)/factorial(n)**2
+
+
+def test__eval_product():
+    from sympy.abc import i, n
+    # issue 4809
+    a = Function('a')
+    assert product(2*a(i), (i, 1, n)) == 2**n * Product(a(i), (i, 1, n))
+    # issue 4810
+    assert product(2**i, (i, 1, n)) == 2**(n*(n + 1)/2)
+    k, m = symbols('k m', integer=True)
+    assert product(2**i, (i, k, m)) == 2**(-k**2/2 + k/2 + m**2/2 + m/2)
+    n = Symbol('n', negative=True, integer=True)
+    p = Symbol('p', positive=True, integer=True)
+    assert product(2**i, (i, n, p)) == 2**(-n**2/2 + n/2 + p**2/2 + p/2)
+    assert product(2**i, (i, p, n)) == 2**(n**2/2 + n/2 - p**2/2 + p/2)
+
+
+def test_product_pow():
+    # issue 4817
+    assert product(2**f(k), (k, 1, n)) == 2**Sum(f(k), (k, 1, n))
+    assert product(2**(2*f(k)), (k, 1, n)) == 2**Sum(2*f(k), (k, 1, n))
+
+
+def test_infinite_product():
+    # issue 5737
+    assert isinstance(Product(2**(1/factorial(n)), (n, 0, oo)), Product)
+
+
+def test_conjugate_transpose():
+    p = Product(x**k, (k, 1, 3))
+    assert p.adjoint().doit() == p.doit().adjoint()
+    assert p.conjugate().doit() == p.doit().conjugate()
+    assert p.transpose().doit() == p.doit().transpose()
+
+    A, B = symbols("A B", commutative=False)
+    p = Product(A*B**k, (k, 1, 3))
+    assert p.adjoint().doit() == p.doit().adjoint()
+    assert p.conjugate().doit() == p.doit().conjugate()
+    assert p.transpose().doit() == p.doit().transpose()
+
+    p = Product(B**k*A, (k, 1, 3))
+    assert p.adjoint().doit() == p.doit().adjoint()
+    assert p.conjugate().doit() == p.doit().conjugate()
+    assert p.transpose().doit() == p.doit().transpose()
+
+
+def test_simplify_prod():
+    y, t, b, c, v, d = symbols('y, t, b, c, v, d', integer = True)
+
+    _simplify = lambda e: simplify(e, doit=False)
+    assert _simplify(Product(x*y, (x, n, m), (y, a, k)) * \
+        Product(y, (x, n, m), (y, a, k))) == \
+            Product(x*y**2, (x, n, m), (y, a, k))
+    assert _simplify(3 * y* Product(x, (x, n, m)) * Product(x, (x, m + 1, a))) \
+        == 3 * y * Product(x, (x, n, a))
+    assert _simplify(Product(x, (x, k + 1, a)) * Product(x, (x, n, k))) == \
+        Product(x, (x, n, a))
+    assert _simplify(Product(x, (x, k + 1, a)) * Product(x + 1, (x, n, k))) == \
+        Product(x, (x, k + 1, a)) * Product(x + 1, (x, n, k))
+    assert _simplify(Product(x, (t, a, b)) * Product(y, (t, a, b)) * \
+        Product(x, (t, b+1, c))) == Product(x*y, (t, a, b)) * \
+            Product(x, (t, b+1, c))
+    assert _simplify(Product(x, (t, a, b)) * Product(x, (t, b+1, c)) * \
+        Product(y, (t, a, b))) == Product(x*y, (t, a, b)) * \
+            Product(x, (t, b+1, c))
+    assert _simplify(Product(sin(t)**2 + cos(t)**2 + 1, (t, a, b))) == \
+        Product(2, (t, a, b))
+    assert _simplify(Product(sin(t)**2 + cos(t)**2 - 1, (t, a, b))) == \
+           Product(0, (t, a, b))
+    assert _simplify(Product(v*Product(sin(t)**2 + cos(t)**2, (t, a, b)),
+                             (v, c, d))) == Product(v*Product(1, (t, a, b)), (v, c, d))
+
+
+def test_change_index():
+    b, y, c, d, z = symbols('b, y, c, d, z', integer = True)
+
+    assert Product(x, (x, a, b)).change_index(x, x + 1, y) == \
+        Product(y - 1, (y, a + 1, b + 1))
+    assert Product(x**2, (x, a, b)).change_index(x, x - 1) == \
+        Product((x + 1)**2, (x, a - 1, b - 1))
+    assert Product(x**2, (x, a, b)).change_index(x, -x, y) == \
+        Product((-y)**2, (y, -b, -a))
+    assert Product(x, (x, a, b)).change_index(x, -x - 1) == \
+        Product(-x - 1, (x, - b - 1, -a - 1))
+    assert Product(x*y, (x, a, b), (y, c, d)).change_index(x, x - 1, z) == \
+        Product((z + 1)*y, (z, a - 1, b - 1), (y, c, d))
+
+
+def test_reorder():
+    b, y, c, d, z = symbols('b, y, c, d, z', integer = True)
+
+    assert Product(x*y, (x, a, b), (y, c, d)).reorder((0, 1)) == \
+        Product(x*y, (y, c, d), (x, a, b))
+    assert Product(x, (x, a, b), (x, c, d)).reorder((0, 1)) == \
+        Product(x, (x, c, d), (x, a, b))
+    assert Product(x*y + z, (x, a, b), (z, m, n), (y, c, d)).reorder(\
+        (2, 0), (0, 1)) == Product(x*y + z, (z, m, n), (y, c, d), (x, a, b))
+    assert Product(x*y*z, (x, a, b), (y, c, d), (z, m, n)).reorder(\
+        (0, 1), (1, 2), (0, 2)) == \
+        Product(x*y*z, (x, a, b), (z, m, n), (y, c, d))
+    assert Product(x*y*z, (x, a, b), (y, c, d), (z, m, n)).reorder(\
+        (x, y), (y, z), (x, z)) == \
+        Product(x*y*z, (x, a, b), (z, m, n), (y, c, d))
+    assert Product(x*y, (x, a, b), (y, c, d)).reorder((x, 1)) == \
+        Product(x*y, (y, c, d), (x, a, b))
+    assert Product(x*y, (x, a, b), (y, c, d)).reorder((y, x)) == \
+        Product(x*y, (y, c, d), (x, a, b))
+
+
+def test_Product_is_convergent():
+    assert Product(1/n**2, (n, 1, oo)).is_convergent() is S.false
+    assert Product(exp(1/n**2), (n, 1, oo)).is_convergent() is S.true
+    assert Product(1/n, (n, 1, oo)).is_convergent() is S.false
+    assert Product(1 + 1/n, (n, 1, oo)).is_convergent() is S.false
+    assert Product(1 + 1/n**2, (n, 1, oo)).is_convergent() is S.true
+
+
+def test_reverse_order():
+    x, y, a, b, c, d= symbols('x, y, a, b, c, d', integer = True)
+
+    assert Product(x, (x, 0, 3)).reverse_order(0) == Product(1/x, (x, 4, -1))
+    assert Product(x*y, (x, 1, 5), (y, 0, 6)).reverse_order(0, 1) == \
+           Product(x*y, (x, 6, 0), (y, 7, -1))
+    assert Product(x, (x, 1, 2)).reverse_order(0) == Product(1/x, (x, 3, 0))
+    assert Product(x, (x, 1, 3)).reverse_order(0) == Product(1/x, (x, 4, 0))
+    assert Product(x, (x, 1, a)).reverse_order(0) == Product(1/x, (x, a + 1, 0))
+    assert Product(x, (x, a, 5)).reverse_order(0) == Product(1/x, (x, 6, a - 1))
+    assert Product(x, (x, a + 1, a + 5)).reverse_order(0) == \
+           Product(1/x, (x, a + 6, a))
+    assert Product(x, (x, a + 1, a + 2)).reverse_order(0) == \
+           Product(1/x, (x, a + 3, a))
+    assert Product(x, (x, a + 1, a + 1)).reverse_order(0) == \
+           Product(1/x, (x, a + 2, a))
+    assert Product(x, (x, a, b)).reverse_order(0) == Product(1/x, (x, b + 1, a - 1))
+    assert Product(x, (x, a, b)).reverse_order(x) == Product(1/x, (x, b + 1, a - 1))
+    assert Product(x*y, (x, a, b), (y, 2, 5)).reverse_order(x, 1) == \
+           Product(x*y, (x, b + 1, a - 1), (y, 6, 1))
+    assert Product(x*y, (x, a, b), (y, 2, 5)).reverse_order(y, x) == \
+           Product(x*y, (x, b + 1, a - 1), (y, 6, 1))
+
+
+def test_issue_9983():
+    n = Symbol('n', integer=True, positive=True)
+    p = Product(1 + 1/n**Rational(2, 3), (n, 1, oo))
+    assert p.is_convergent() is S.false
+    assert product(1 + 1/n**Rational(2, 3), (n, 1, oo)) == p.doit()
+
+
+def test_issue_13546():
+    n = Symbol('n')
+    k = Symbol('k')
+    p = Product(n + 1 / 2**k, (k, 0, n-1)).doit()
+    assert p.subs(n, 2).doit() == Rational(15, 2)
+
+
+def test_issue_14036():
+    a, n = symbols('a n')
+    assert product(1 - a**2 / (n*pi)**2, [n, 1, oo]) != 0
+
+
+def test_rewrite_Sum():
+    assert Product(1 - S.Half**2/k**2, (k, 1, oo)).rewrite(Sum) == \
+        exp(Sum(log(1 - 1/(4*k**2)), (k, 1, oo)))
+
+
+def test_KroneckerDelta_Product():
+    y = Symbol('y')
+    assert Product(x*KroneckerDelta(x, y), (x, 0, 1)).doit() == 0
+
+
+def test_issue_20848():
+    _i = Dummy('i')
+    t, y, z = symbols('t y z')
+    assert diff(Product(x, (y, 1, z)), x).as_dummy() == Sum(Product(x, (y, 1, _i - 1))*Product(x, (y, _i + 1, z)), (_i, 1, z)).as_dummy()
+    assert diff(Product(x, (y, 1, z)), x).doit() == x**(z - 1)*z
+    assert diff(Product(x, (y, x, z)), x) == Derivative(Product(x, (y, x, z)), x)
+    assert diff(Product(t, (x, 1, z)), x) == S(0)
+    assert Product(sin(n*x), (n, -1, 1)).diff(x).doit() == S(0)

.venv/lib/python3.13/site-packages/sympy/concrete/tests/test_sums_products.py

@@ -0,0 +1,1676 @@
+from math import prod
+
+from sympy.concrete.expr_with_intlimits import ReorderError
+from sympy.concrete.products import (Product, product)
+from sympy.concrete.summations import (Sum, summation, telescopic,
+     eval_sum_residue, _dummy_with_inherited_properties_concrete)
+from sympy.core.function import (Derivative, Function)
+from sympy.core import (Catalan, EulerGamma)
+from sympy.core.facts import InconsistentAssumptions
+from sympy.core.mod import Mod
+from sympy.core.numbers import (E, I, Rational, nan, oo, pi)
+from sympy.core.relational import Eq, Ne
+from sympy.core.numbers import Float
+from sympy.core.singleton import S
+from sympy.core.symbol import (Dummy, Symbol, symbols)
+from sympy.core.sympify import sympify
+from sympy.functions.combinatorial.factorials import (rf, binomial, factorial)
+from sympy.functions.combinatorial.numbers import harmonic
+from sympy.functions.elementary.complexes import Abs, re
+from sympy.functions.elementary.exponential import (exp, log)
+from sympy.functions.elementary.hyperbolic import (sinh, tanh)
+from sympy.functions.elementary.integers import floor
+from sympy.functions.elementary.miscellaneous import sqrt
+from sympy.functions.elementary.piecewise import Piecewise
+from sympy.functions.elementary.trigonometric import (cos, sin, atan)
+from sympy.functions.special.gamma_functions import (gamma, lowergamma)
+from sympy.functions.special.tensor_functions import KroneckerDelta
+from sympy.functions.special.zeta_functions import zeta
+from sympy.integrals.integrals import Integral
+from sympy.logic.boolalg import And, Or
+from sympy.matrices.expressions.matexpr import MatrixSymbol
+from sympy.matrices.expressions.special import Identity
+from sympy.matrices import (Matrix, SparseMatrix,
+    ImmutableDenseMatrix, ImmutableSparseMatrix, diag)
+from sympy.sets.contains import Contains
+from sympy.sets.fancysets import Range
+from sympy.sets.sets import Interval
+from sympy.simplify.combsimp import combsimp
+from sympy.simplify.simplify import simplify
+from sympy.tensor.indexed import (Idx, Indexed, IndexedBase)
+from sympy.testing.pytest import XFAIL, raises, slow
+from sympy.abc import a, b, c, d, k, m, x, y, z
+
+n = Symbol('n', integer=True)
+f, g = symbols('f g', cls=Function)
+
+def test_karr_convention():
+    # Test the Karr summation convention that we want to hold.
+    # See his paper "Summation in Finite Terms" for a detailed
+    # reasoning why we really want exactly this definition.
+    # The convention is described on page 309 and essentially
+    # in section 1.4, definition 3:
+    #
+    # \sum_{m <= i < n} f(i) 'has the obvious meaning'   for m < n
+    # \sum_{m <= i < n} f(i) = 0                         for m = n
+    # \sum_{m <= i < n} f(i) = - \sum_{n <= i < m} f(i)  for m > n
+    #
+    # It is important to note that he defines all sums with
+    # the upper limit being *exclusive*.
+    # In contrast, SymPy and the usual mathematical notation has:
+    #
+    # sum_{i = a}^b f(i) = f(a) + f(a+1) + ... + f(b-1) + f(b)
+    #
+    # with the upper limit *inclusive*. So translating between
+    # the two we find that:
+    #
+    # \sum_{m <= i < n} f(i) = \sum_{i = m}^{n-1} f(i)
+    #
+    # where we intentionally used two different ways to typeset the
+    # sum and its limits.
+
+    i = Symbol("i", integer=True)
+    k = Symbol("k", integer=True)
+    j = Symbol("j", integer=True)
+
+    # A simple example with a concrete summand and symbolic limits.
+
+    # The normal sum: m = k and n = k + j and therefore m < n:
+    m = k
+    n = k + j
+
+    a = m
+    b = n - 1
+    S1 = Sum(i**2, (i, a, b)).doit()
+
+    # The reversed sum: m = k + j and n = k and therefore m > n:
+    m = k + j
+    n = k
+
+    a = m
+    b = n - 1
+    S2 = Sum(i**2, (i, a, b)).doit()
+
+    assert simplify(S1 + S2) == 0
+
+    # Test the empty sum: m = k and n = k and therefore m = n:
+    m = k
+    n = k
+
+    a = m
+    b = n - 1
+    Sz = Sum(i**2, (i, a, b)).doit()
+
+    assert Sz == 0
+
+    # Another example this time with an unspecified summand and
+    # numeric limits. (We can not do both tests in the same example.)
+
+    # The normal sum with m < n:
+    m = 2
+    n = 11
+
+    a = m
+    b = n - 1
+    S1 = Sum(f(i), (i, a, b)).doit()
+
+    # The reversed sum with m > n:
+    m = 11
+    n = 2
+
+    a = m
+    b = n - 1
+    S2 = Sum(f(i), (i, a, b)).doit()
+
+    assert simplify(S1 + S2) == 0
+
+    # Test the empty sum with m = n:
+    m = 5
+    n = 5
+
+    a = m
+    b = n - 1
+    Sz = Sum(f(i), (i, a, b)).doit()
+
+    assert Sz == 0
+
+    e = Piecewise((exp(-i), Mod(i, 2) > 0), (0, True))
+    s = Sum(e, (i, 0, 11))
+    assert s.n(3) == s.doit().n(3)
+
+    # issue #27893
+    n = Symbol('n', integer=True)
+    assert Sum(1/(x**2 + 1), (x, oo, 0)).doit(deep=False) == Rational(-1, 2) + pi / (2 * tanh(pi))
+    assert Sum(c**x/factorial(x), (x, oo, 0)).doit(deep=False).simplify() == exp(c) - 1 # exponential series
+    assert Sum((-1)**x/x, (x, oo,0)).doit() == -log(2) # alternating harmnic series
+    assert Sum((1/2)**x,(x, oo, -1)).doit() == S(2) # geometric series
+    assert Sum(1/x, (x, oo, 0)).doit() == oo # harmonic series, divergent
+    assert Sum((-1)**x/(2*x+1), (x, oo, -1)).doit() == pi/4 # leibniz series
+    assert Sum((((-1)**x) * c**(2*x+1)) / factorial(2*x+1), (x, oo, -1)).doit() == sin(c) # sinusoidal series
+    assert Sum((((-1)**x) * c**(2*x+1)) / (2*x+1), (x, 0, oo)).doit() \
+        == Piecewise((atan(c), Ne(c**2, -1) & (Abs(c**2) <= 1)), \
+                     (Sum((-1)**x*c**(2*x + 1)/(2*x + 1), (x, 0, oo)), True)) # arctangent series
+    assert Sum(binomial(n, x) * c**x, (x, 0, oo)).doit() \
+        == Piecewise(((c + 1)**n, \
+                     ((n <= -1) & (Abs(c) < 1)) \
+                        | ((n > 0) & (Abs(c) <= 1)) \
+                        | ((n <= 0) & (n > -1) & Ne(c, -1) & (Abs(c) <= 1))), \
+                     (Sum(c**x*binomial(n, x), (x, 0, oo)), True)) # binomial series
+    assert Sum(1/x**n, (x, oo, 0)).doit() \
+        == Piecewise((zeta(n), n > 1), (Sum(x**(-n), (x, oo, 0)), True)) # Euler's zeta function
+
+def test_karr_proposition_2a():
+    # Test Karr, page 309, proposition 2, part a
+    i = Symbol("i", integer=True)
+    u = Symbol("u", integer=True)
+    v = Symbol("v", integer=True)
+
+    def test_the_sum(m, n):
+        # g
+        g = i**3 + 2*i**2 - 3*i
+        # f = Delta g
+        f = simplify(g.subs(i, i+1) - g)
+        # The sum
+        a = m
+        b = n - 1
+        S = Sum(f, (i, a, b)).doit()
+        # Test if Sum_{m <= i < n} f(i) = g(n) - g(m)
+        assert simplify(S - (g.subs(i, n) - g.subs(i, m))) == 0
+
+    # m < n
+    test_the_sum(u,   u+v)
+    # m = n
+    test_the_sum(u,   u  )
+    # m > n
+    test_the_sum(u+v, u  )
+
+
+def test_karr_proposition_2b():
+    # Test Karr, page 309, proposition 2, part b
+    i = Symbol("i", integer=True)
+    u = Symbol("u", integer=True)
+    v = Symbol("v", integer=True)
+    w = Symbol("w", integer=True)
+
+    def test_the_sum(l, n, m):
+        # Summand
+        s = i**3
+        # First sum
+        a = l
+        b = n - 1
+        S1 = Sum(s, (i, a, b)).doit()
+        # Second sum
+        a = l
+        b = m - 1
+        S2 = Sum(s, (i, a, b)).doit()
+        # Third sum
+        a = m
+        b = n - 1
+        S3 = Sum(s, (i, a, b)).doit()
+        # Test if S1 = S2 + S3 as required
+        assert S1 - (S2 + S3) == 0
+
+    # l < m < n
+    test_the_sum(u,     u+v,   u+v+w)
+    # l < m = n
+    test_the_sum(u,     u+v,   u+v  )
+    # l < m > n
+    test_the_sum(u,     u+v+w, v    )
+    # l = m < n
+    test_the_sum(u,     u,     u+v  )
+    # l = m = n
+    test_the_sum(u,     u,     u    )
+    # l = m > n
+    test_the_sum(u+v,   u+v,   u    )
+    # l > m < n
+    test_the_sum(u+v,   u,     u+w  )
+    # l > m = n
+    test_the_sum(u+v,   u,     u    )
+    # l > m > n
+    test_the_sum(u+v+w, u+v,   u    )
+
+
+def test_arithmetic_sums():
+    assert summation(1, (n, a, b)) == b - a + 1
+    assert Sum(S.NaN, (n, a, b)) is S.NaN
+    assert Sum(x, (n, a, a)).doit() == x
+    assert Sum(x, (x, a, a)).doit() == a
+    assert Sum(x, (n, 1, a)).doit() == a*x
+    assert Sum(x, (x, Range(1, 11))).doit() == 55
+    assert Sum(x, (x, Range(1, 11, 2))).doit() == 25
+    assert Sum(x, (x, Range(1, 10, 2))) == Sum(x, (x, Range(9, 0, -2)))
+    lo, hi = 1, 2
+    s1 = Sum(n, (n, lo, hi))
+    s2 = Sum(n, (n, hi, lo))
+    assert s1 != s2
+    assert s1.doit() == 3 and s2.doit() == 0
+    lo, hi = x, x + 1
+    s1 = Sum(n, (n, lo, hi))
+    s2 = Sum(n, (n, hi, lo))
+    assert s1 != s2
+    assert s1.doit() == 2*x + 1 and s2.doit() == 0
+    assert Sum(Integral(x, (x, 1, y)) + x, (x, 1, 2)).doit() == \
+        y**2 + 2
+    assert summation(1, (n, 1, 10)) == 10
+    assert summation(2*n, (n, 0, 10**10)) == 100000000010000000000
+    assert summation(4*n*m, (n, a, 1), (m, 1, d)).expand() == \
+        2*d + 2*d**2 + a*d + a*d**2 - d*a**2 - a**2*d**2
+    assert summation(cos(n), (n, -2, 1)) == cos(-2) + cos(-1) + cos(0) + cos(1)
+    assert summation(cos(n), (n, x, x + 2)) == cos(x) + cos(x + 1) + cos(x + 2)
+    assert isinstance(summation(cos(n), (n, x, x + S.Half)), Sum)
+    assert summation(k, (k, 0, oo)) is oo
+    assert summation(k, (k, Range(1, 11))) == 55
+
+
+def test_polynomial_sums():
+    assert summation(n**2, (n, 3, 8)) == 199
+    assert summation(n, (n, a, b)) == \
+        ((a + b)*(b - a + 1)/2).expand()
+    assert summation(n**2, (n, 1, b)) == \
+        ((2*b**3 + 3*b**2 + b)/6).expand()
+    assert summation(n**3, (n, 1, b)) == \
+        ((b**4 + 2*b**3 + b**2)/4).expand()
+    assert summation(n**6, (n, 1, b)) == \
+        ((6*b**7 + 21*b**6 + 21*b**5 - 7*b**3 + b)/42).expand()
+
+
+def test_geometric_sums():
+    assert summation(pi**n, (n, 0, b)) == (1 - pi**(b + 1)) / (1 - pi)
+    assert summation(2 * 3**n, (n, 0, b)) == 3**(b + 1) - 1
+    assert summation(S.Half**n, (n, 1, oo)) == 1
+    assert summation(2**n, (n, 0, b)) == 2**(b + 1) - 1
+    assert summation(2**n, (n, 1, oo)) is oo
+    assert summation(2**(-n), (n, 1, oo)) == 1
+    assert summation(3**(-n), (n, 4, oo)) == Rational(1, 54)
+    assert summation(2**(-4*n + 3), (n, 1, oo)) == Rational(8, 15)
+    assert summation(2**(n + 1), (n, 1, b)).expand() == 4*(2**b - 1)
+
+    # issue 6664:
+    assert summation(x**n, (n, 0, oo)) == \
+        Piecewise((1/(-x + 1), Abs(x) < 1), (Sum(x**n, (n, 0, oo)), True))
+
+    assert summation(-2**n, (n, 0, oo)) is -oo
+    assert summation(I**n, (n, 0, oo)) == Sum(I**n, (n, 0, oo))
+
+    # issue 6802:
+    assert summation((-1)**(2*x + 2), (x, 0, n)) == n + 1
+    assert summation((-2)**(2*x + 2), (x, 0, n)) == 4*4**(n + 1)/S(3) - Rational(4, 3)
+    assert summation((-1)**x, (x, 0, n)) == -(-1)**(n + 1)/S(2) + S.Half
+    assert summation(y**x, (x, a, b)) == \
+        Piecewise((-a + b + 1, Eq(y, 1)), ((y**a - y**(b + 1))/(-y + 1), True))
+    assert summation((-2)**(y*x + 2), (x, 0, n)) == \
+        4*Piecewise((n + 1, Eq((-2)**y, 1)),
+                    ((-(-2)**(y*(n + 1)) + 1)/(-(-2)**y + 1), True))
+
+    # issue 8251:
+    assert summation((1/(n + 1)**2)*n**2, (n, 0, oo)) is oo
+
+    #issue 9908:
+    assert Sum(1/(n**3 - 1), (n, -oo, -2)).doit() == summation(1/(n**3 - 1), (n, -oo, -2))
+
+    #issue 11642:
+    result = Sum(0.5**n, (n, 1, oo)).doit()
+    assert result == 1.0
+    assert result.is_Float
+
+    result = Sum(0.25**n, (n, 1, oo)).doit()
+    assert result == 1/3.
+    assert result.is_Float
+
+    result = Sum(0.99999**n, (n, 1, oo)).doit()
+    assert result == 99999.0
+    assert result.is_Float
+
+    result = Sum(S.Half**n, (n, 1, oo)).doit()
+    assert result == 1
+    assert not result.is_Float
+
+    result = Sum(Rational(3, 5)**n, (n, 1, oo)).doit()
+    assert result == Rational(3, 2)
+    assert not result.is_Float
+
+    assert Sum(1.0**n, (n, 1, oo)).doit() is oo
+    assert Sum(2.43**n, (n, 1, oo)).doit() is oo
+
+    # Issue 13979
+    i, k, q = symbols('i k q', integer=True)
+    result = summation(
+        exp(-2*I*pi*k*i/n) * exp(2*I*pi*q*i/n) / n, (i, 0, n - 1)
+    )
+    assert result.simplify() == Piecewise(
+            (1, Eq(exp(-2*I*pi*(k - q)/n), 1)), (0, True)
+    )
+
+    #Issue 23491
+    assert Sum(1/(n**2 + 1), (n, 1, oo)).doit() == S(-1)/2 + pi/(2*tanh(pi))
+
+def test_harmonic_sums():
+    assert summation(1/k, (k, 0, n)) == Sum(1/k, (k, 0, n))
+    assert summation(1/k, (k, 1, n)) == harmonic(n)
+    assert summation(n/k, (k, 1, n)) == n*harmonic(n)
+    assert summation(1/k, (k, 5, n)) == harmonic(n) - harmonic(4)
+
+
+def test_composite_sums():
+    f = S.Half*(7 - 6*n + Rational(1, 7)*n**3)
+    s = summation(f, (n, a, b))
+    assert not isinstance(s, Sum)
+    A = 0
+    for i in range(-3, 5):
+        A += f.subs(n, i)
+    B = s.subs(a, -3).subs(b, 4)
+    assert A == B
+
+
+def test_hypergeometric_sums():
+    assert summation(
+        binomial(2*k, k)/4**k, (k, 0, n)) == (1 + 2*n)*binomial(2*n, n)/4**n
+    assert summation(binomial(2*k, k)/5**k, (k, -oo, oo)) == sqrt(5)
+
+
+def test_other_sums():
+    f = m**2 + m*exp(m)
+    g = 3*exp(Rational(3, 2))/2 + exp(S.Half)/2 - exp(Rational(-1, 2))/2 - 3*exp(Rational(-3, 2))/2 + 5
+
+    assert summation(f, (m, Rational(-3, 2), Rational(3, 2))) == g
+    assert summation(f, (m, -1.5, 1.5)).evalf().epsilon_eq(g.evalf(), 1e-10)
+
+fac = factorial
+
+
+def NS(e, n=15, **options):
+    return str(sympify(e).evalf(n, **options))
+
+
+def test_evalf_fast_series():
+    # Euler transformed series for sqrt(1+x)
+    assert NS(Sum(
+        fac(2*n + 1)/fac(n)**2/2**(3*n + 1), (n, 0, oo)), 100) == NS(sqrt(2), 100)
+
+    # Some series for exp(1)
+    estr = NS(E, 100)
+    assert NS(Sum(1/fac(n), (n, 0, oo)), 100) == estr
+    assert NS(1/Sum((1 - 2*n)/fac(2*n), (n, 0, oo)), 100) == estr
+    assert NS(Sum((2*n + 1)/fac(2*n), (n, 0, oo)), 100) == estr
+    assert NS(Sum((4*n + 3)/2**(2*n + 1)/fac(2*n + 1), (n, 0, oo))**2, 100) == estr
+
+    pistr = NS(pi, 100)
+    # Ramanujan series for pi
+    assert NS(9801/sqrt(8)/Sum(fac(
+        4*n)*(1103 + 26390*n)/fac(n)**4/396**(4*n), (n, 0, oo)), 100) == pistr
+    assert NS(1/Sum(
+        binomial(2*n, n)**3 * (42*n + 5)/2**(12*n + 4), (n, 0, oo)), 100) == pistr
+    # Machin's formula for pi
+    assert NS(16*Sum((-1)**n/(2*n + 1)/5**(2*n + 1), (n, 0, oo)) -
+        4*Sum((-1)**n/(2*n + 1)/239**(2*n + 1), (n, 0, oo)), 100) == pistr
+
+    # Apery's constant
+    astr = NS(zeta(3), 100)
+    P = 126392*n**5 + 412708*n**4 + 531578*n**3 + 336367*n**2 + 104000* \
+        n + 12463
+    assert NS(Sum((-1)**n * P / 24 * (fac(2*n + 1)*fac(2*n)*fac(
+        n))**3 / fac(3*n + 2) / fac(4*n + 3)**3, (n, 0, oo)), 100) == astr
+    assert NS(Sum((-1)**n * (205*n**2 + 250*n + 77)/64 * fac(n)**10 /
+              fac(2*n + 1)**5, (n, 0, oo)), 100) == astr
+
+
+def test_evalf_fast_series_issue_4021():
+    # Catalan's constant
+    assert NS(Sum((-1)**(n - 1)*2**(8*n)*(40*n**2 - 24*n + 3)*fac(2*n)**3*
+        fac(n)**2/n**3/(2*n - 1)/fac(4*n)**2, (n, 1, oo))/64, 100) == \
+        NS(Catalan, 100)
+    astr = NS(zeta(3), 100)
+    assert NS(5*Sum(
+        (-1)**(n - 1)*fac(n)**2 / n**3 / fac(2*n), (n, 1, oo))/2, 100) == astr
+    assert NS(Sum((-1)**(n - 1)*(56*n**2 - 32*n + 5) / (2*n - 1)**2 * fac(n - 1)
+              **3 / fac(3*n), (n, 1, oo))/4, 100) == astr
+
+
+def test_evalf_slow_series():
+    assert NS(Sum((-1)**n / n, (n, 1, oo)), 15) == NS(-log(2), 15)
+    assert NS(Sum((-1)**n / n, (n, 1, oo)), 50) == NS(-log(2), 50)
+    assert NS(Sum(1/n**2, (n, 1, oo)), 15) == NS(pi**2/6, 15)
+    assert NS(Sum(1/n**2, (n, 1, oo)), 100) == NS(pi**2/6, 100)
+    assert NS(Sum(1/n**2, (n, 1, oo)), 500) == NS(pi**2/6, 500)
+    assert NS(Sum((-1)**n / (2*n + 1)**3, (n, 0, oo)), 15) == NS(pi**3/32, 15)
+    assert NS(Sum((-1)**n / (2*n + 1)**3, (n, 0, oo)), 50) == NS(pi**3/32, 50)
+
+
+def test_evalf_oo_to_oo():
+    # There used to be an error in certain cases
+    # Does not evaluate, but at least do not throw an error
+    # Evaluates symbolically to 0, which is not correct
+    assert Sum(1/(n**2+1), (n, -oo, oo)).evalf() == Sum(1/(n**2+1), (n, -oo, oo))
+    # This evaluates if from 1 to oo and symbolically
+    assert Sum(1/(factorial(abs(n))), (n, -oo, -1)).evalf() == Sum(1/(factorial(abs(n))), (n, -oo, -1))
+
+
+def test_euler_maclaurin():
+    # Exact polynomial sums with E-M
+    def check_exact(f, a, b, m, n):
+        A = Sum(f, (k, a, b))
+        s, e = A.euler_maclaurin(m, n)
+        assert (e == 0) and (s.expand() == A.doit())
+    check_exact(k**4, a, b, 0, 2)
+    check_exact(k**4 + 2*k, a, b, 1, 2)
+    check_exact(k**4 + k**2, a, b, 1, 5)
+    check_exact(k**5, 2, 6, 1, 2)
+    check_exact(k**5, 2, 6, 1, 3)
+    assert Sum(x-1, (x, 0, 2)).euler_maclaurin(m=30, n=30, eps=2**-15) == (0, 0)
+    # Not exact
+    assert Sum(k**6, (k, a, b)).euler_maclaurin(0, 2)[1] != 0
+    # Numerical test
+    for mi, ni in [(2, 4), (2, 20), (10, 20), (18, 20)]:
+        A = Sum(1/k**3, (k, 1, oo))
+        s, e = A.euler_maclaurin(mi, ni)
+        assert abs((s - zeta(3)).evalf()) < e.evalf()
+
+    raises(ValueError, lambda: Sum(1, (x, 0, 1), (k, 0, 1)).euler_maclaurin())
+
+
+@slow
+def test_evalf_euler_maclaurin():
+    assert NS(Sum(1/k**k, (k, 1, oo)), 15) == '1.29128599706266'
+    assert NS(Sum(1/k**k, (k, 1, oo)),
+              50) == '1.2912859970626635404072825905956005414986193682745'
+    assert NS(Sum(1/k - log(1 + 1/k), (k, 1, oo)), 15) == NS(EulerGamma, 15)
+    assert NS(Sum(1/k - log(1 + 1/k), (k, 1, oo)), 50) == NS(EulerGamma, 50)
+    assert NS(Sum(log(k)/k**2, (k, 1, oo)), 15) == '0.937548254315844'
+    assert NS(Sum(log(k)/k**2, (k, 1, oo)),
+              50) == '0.93754825431584375370257409456786497789786028861483'
+    assert NS(Sum(1/k, (k, 1000000, 2000000)), 15) == '0.693147930560008'
+    assert NS(Sum(1/k, (k, 1000000, 2000000)),
+              50) == '0.69314793056000780941723211364567656807940638436025'
+
+
+def test_evalf_symbolic():
+    # issue 6328
+    expr = Sum(f(x), (x, 1, 3)) + Sum(g(x), (x, 1, 3))
+    assert expr.evalf() == expr
+
+
+def test_evalf_issue_3273():
+    assert Sum(0, (k, 1, oo)).evalf() == 0
+
+
+def test_simple_products():
+    assert Product(S.NaN, (x, 1, 3)) is S.NaN
+    assert product(S.NaN, (x, 1, 3)) is S.NaN
+    assert Product(x, (n, a, a)).doit() == x
+    assert Product(x, (x, a, a)).doit() == a
+    assert Product(x, (y, 1, a)).doit() == x**a
+
+    lo, hi = 1, 2
+    s1 = Product(n, (n, lo, hi))
+    s2 = Product(n, (n, hi, lo))
+    assert s1 != s2
+    # This IS correct according to Karr product convention
+    assert s1.doit() == 2
+    assert s2.doit() == 1
+
+    lo, hi = x, x + 1
+    s1 = Product(n, (n, lo, hi))
+    s2 = Product(n, (n, hi, lo))
+    s3 = 1 / Product(n, (n, hi + 1, lo - 1))
+    assert s1 != s2
+    # This IS correct according to Karr product convention
+    assert s1.doit() == x*(x + 1)
+    assert s2.doit() == 1
+    assert s3.doit() == x*(x + 1)
+
+    assert Product(Integral(2*x, (x, 1, y)) + 2*x, (x, 1, 2)).doit() == \
+        (y**2 + 1)*(y**2 + 3)
+    assert product(2, (n, a, b)) == 2**(b - a + 1)
+    assert product(n, (n, 1, b)) == factorial(b)
+    assert product(n**3, (n, 1, b)) == factorial(b)**3
+    assert product(3**(2 + n), (n, a, b)) \
+        == 3**(2*(1 - a + b) + b/2 + (b**2)/2 + a/2 - (a**2)/2)
+    assert product(cos(n), (n, 3, 5)) == cos(3)*cos(4)*cos(5)
+    assert product(cos(n), (n, x, x + 2)) == cos(x)*cos(x + 1)*cos(x + 2)
+    assert isinstance(product(cos(n), (n, x, x + S.Half)), Product)
+    # If Product managed to evaluate this one, it most likely got it wrong!
+    assert isinstance(Product(n**n, (n, 1, b)), Product)
+
+
+def test_rational_products():
+    assert combsimp(product(1 + 1/n, (n, a, b))) == (1 + b)/a
+    assert combsimp(product(n + 1, (n, a, b))) == gamma(2 + b)/gamma(1 + a)
+    assert combsimp(product((n + 1)/(n - 1), (n, a, b))) == b*(1 + b)/(a*(a - 1))
+    assert combsimp(product(n/(n + 1)/(n + 2), (n, a, b))) == \
+        a*gamma(a + 2)/(b + 1)/gamma(b + 3)
+    assert combsimp(product(n*(n + 1)/(n - 1)/(n - 2), (n, a, b))) == \
+        b**2*(b - 1)*(1 + b)/(a - 1)**2/(a*(a - 2))
+
+
+def test_wallis_product():
+    # Wallis product, given in two different forms to ensure that Product
+    # can factor simple rational expressions
+    A = Product(4*n**2 / (4*n**2 - 1), (n, 1, b))
+    B = Product((2*n)*(2*n)/(2*n - 1)/(2*n + 1), (n, 1, b))
+    R = pi*gamma(b + 1)**2/(2*gamma(b + S.Half)*gamma(b + Rational(3, 2)))
+    assert simplify(A.doit()) == R
+    assert simplify(B.doit()) == R
+    # This one should eventually also be doable (Euler's product formula for sin)
+    # assert Product(1+x/n**2, (n, 1, b)) == ...
+
+
+def test_telescopic_sums():
+    #checks also input 2 of comment 1 issue 4127
+    assert Sum(1/k - 1/(k + 1), (k, 1, n)).doit() == 1 - 1/(1 + n)
+    assert Sum(
+        f(k) - f(k + 2), (k, m, n)).doit() == -f(1 + n) - f(2 + n) + f(m) + f(1 + m)
+    assert Sum(cos(k) - cos(k + 3), (k, 1, n)).doit() == -cos(1 + n) - \
+        cos(2 + n) - cos(3 + n) + cos(1) + cos(2) + cos(3)
+
+    # dummy variable shouldn't matter
+    assert telescopic(1/m, -m/(1 + m), (m, n - 1, n)) == \
+        telescopic(1/k, -k/(1 + k), (k, n - 1, n))
+
+    assert Sum(1/x/(x - 1), (x, a, b)).doit() == 1/(a - 1) - 1/b
+    eq = 1/((5*n + 2)*(5*(n + 1) + 2))
+    assert Sum(eq, (n, 0, oo)).doit() == S(1)/10
+    nz = symbols('nz', nonzero=True)
+    v = Sum(eq.subs(5, nz), (n, 0, oo)).doit()
+    assert v.subs(nz, 5).simplify() == S(1)/10
+    # check that apart is being used in non-symbolic case
+    s = Sum(eq, (n, 0, k)).doit()
+    v = Sum(eq, (n, 0, 10**100)).doit()
+    assert v == s.subs(k, 10**100)
+
+
+def test_sum_reconstruct():
+    s = Sum(n**2, (n, -1, 1))
+    assert s == Sum(*s.args)
+    raises(ValueError, lambda: Sum(x, x))
+    raises(ValueError, lambda: Sum(x, (x, 1)))
+
+
+def test_limit_subs():
+    for F in (Sum, Product, Integral):
+        assert F(a*exp(a), (a, -2, 2)) == F(a*exp(a), (a, -b, b)).subs(b, 2)
+        assert F(a, (a, F(b, (b, 1, 2)), 4)).subs(F(b, (b, 1, 2)), c) == \
+            F(a, (a, c, 4))
+        assert F(x, (x, 1, x + y)).subs(x, 1) == F(x, (x, 1, y + 1))
+
+
+def test_function_subs():
+    S = Sum(x*f(y),(x,0,oo),(y,0,oo))
+    assert S.subs(f(y),y) == Sum(x*y,(x,0,oo),(y,0,oo))
+    assert S.subs(f(x),x) == S
+    raises(ValueError, lambda: S.subs(f(y),x+y) )
+    S = Sum(x*log(y),(x,0,oo),(y,0,oo))
+    assert S.subs(log(y),y) == S
+    S = Sum(x*f(y),(x,0,oo),(y,0,oo))
+    assert S.subs(f(y),y) == Sum(x*y,(x,0,oo),(y,0,oo))
+
+
+def test_equality():
+    # if this fails remove special handling below
+    raises(ValueError, lambda: Sum(x, x))
+    r = symbols('x', real=True)
+    for F in (Sum, Product, Integral):
+        try:
+            assert F(x, x) != F(y, y)
+            assert F(x, (x, 1, 2)) != F(x, x)
+            assert F(x, (x, x)) != F(x, x)  # or else they print the same
+            assert F(1, x) != F(1, y)
+        except ValueError:
+            pass
+        assert F(a, (x, 1, 2)) != F(a, (x, 1, 3))  # diff limit
+        assert F(a, (x, 1, x)) != F(a, (y, 1, y))
+        assert F(a, (x, 1, 2)) != F(b, (x, 1, 2))  # diff expression
+        assert F(x, (x, 1, 2)) != F(r, (r, 1, 2))  # diff assumptions
+        assert F(1, (x, 1, x)) != F(1, (y, 1, x))  # only dummy is diff
+        assert F(1, (x, 1, x)).dummy_eq(F(1, (y, 1, x)))
+
+    # issue 5265
+    assert Sum(x, (x, 1, x)).subs(x, a) == Sum(x, (x, 1, a))
+
+
+def test_Sum_doit():
+    assert Sum(n*Integral(a**2), (n, 0, 2)).doit() == a**3
+    assert Sum(n*Integral(a**2), (n, 0, 2)).doit(deep=False) == \
+        3*Integral(a**2)
+    assert summation(n*Integral(a**2), (n, 0, 2)) == 3*Integral(a**2)
+
+    # test nested sum evaluation
+    s = Sum( Sum( Sum(2,(z,1,n+1)), (y,x+1,n)), (x,1,n))
+    assert 0 == (s.doit() - n*(n+1)*(n-1)).factor()
+
+    # Integer assumes finite
+    assert Sum(KroneckerDelta(x, y), (x, -oo, oo)).doit() == Piecewise((1, And(-oo < y, y < oo)), (0, True))
+    assert Sum(KroneckerDelta(m, n), (m, -oo, oo)).doit() == 1
+    assert Sum(m*KroneckerDelta(x, y), (x, -oo, oo)).doit() == Piecewise((m, And(-oo < y, y < oo)), (0, True))
+    assert Sum(x*KroneckerDelta(m, n), (m, -oo, oo)).doit() == x
+    assert Sum(Sum(KroneckerDelta(m, n), (m, 1, 3)), (n, 1, 3)).doit() == 3
+    assert Sum(Sum(KroneckerDelta(k, m), (m, 1, 3)), (n, 1, 3)).doit() == \
+           3 * Piecewise((1, And(1 <= k, k <= 3)), (0, True))
+    assert Sum(f(n) * Sum(KroneckerDelta(m, n), (m, 0, oo)), (n, 1, 3)).doit() == \
+           f(1) + f(2) + f(3)
+    assert Sum(f(n) * Sum(KroneckerDelta(m, n), (m, 0, oo)), (n, 1, oo)).doit() == \
+           Sum(f(n), (n, 1, oo))
+
+    # issue 2597
+    nmax = symbols('N', integer=True, positive=True)
+    pw = Piecewise((1, And(1 <= n, n <= nmax)), (0, True))
+    assert Sum(pw, (n, 1, nmax)).doit() == Sum(Piecewise((1, nmax >= n),
+                    (0, True)), (n, 1, nmax))
+
+    q, s = symbols('q, s')
+    assert summation(1/n**(2*s), (n, 1, oo)) == Piecewise((zeta(2*s), 2*re(s) > 1),
+        (Sum(n**(-2*s), (n, 1, oo)), True))
+    assert summation(1/(n+1)**s, (n, 0, oo)) == Piecewise((zeta(s), re(s) > 1),
+        (Sum((n + 1)**(-s), (n, 0, oo)), True))
+    assert summation(1/(n+q)**s, (n, 0, oo)) == Piecewise(
+        (zeta(s, q), And(~Contains(-q, S.Naturals0), re(s) > 1)),
+        (Sum((n + q)**(-s), (n, 0, oo)), True))
+    assert summation(1/(n+q)**s, (n, q, oo)) == Piecewise(
+        (zeta(s, 2*q), And(~Contains(-2*q, S.Naturals0), re(s) > 1)),
+        (Sum((n + q)**(-s), (n, q, oo)), True))
+    assert summation(1/n**2, (n, 1, oo)) == zeta(2)
+    assert summation(1/n**s, (n, 0, oo)) == Sum(n**(-s), (n, 0, oo))
+    assert summation(1/(n+1)**(2+I), (n, 0, oo)) == zeta(2+I)
+    t = symbols('t', real=True, positive=True)
+    assert summation(1/(n+I)**(t+1), (n, 0, oo)) == zeta(t+1, I)
+
+
+def test_Product_doit():
+    assert Product(n*Integral(a**2), (n, 1, 3)).doit() == 2 * a**9 / 9
+    assert Product(n*Integral(a**2), (n, 1, 3)).doit(deep=False) == \
+        6*Integral(a**2)**3
+    assert product(n*Integral(a**2), (n, 1, 3)) == 6*Integral(a**2)**3
+
+
+def test_Sum_interface():
+    assert isinstance(Sum(0, (n, 0, 2)), Sum)
+    assert Sum(nan, (n, 0, 2)) is nan
+    assert Sum(nan, (n, 0, oo)) is nan
+    assert Sum(0, (n, 0, 2)).doit() == 0
+    assert isinstance(Sum(0, (n, 0, oo)), Sum)
+    assert Sum(0, (n, 0, oo)).doit() == 0
+    raises(ValueError, lambda: Sum(1))
+    raises(ValueError, lambda: summation(1))
+
+
+def test_diff():
+    assert Sum(x, (x, 1, 2)).diff(x) == 0
+    assert Sum(x*y, (x, 1, 2)).diff(x) == 0
+    assert Sum(x*y, (y, 1, 2)).diff(x) == Sum(y, (y, 1, 2))
+    e = Sum(x*y, (x, 1, a))
+    assert e.diff(a) == Derivative(e, a)
+    assert Sum(x*y, (x, 1, 3), (a, 2, 5)).diff(y).doit() == \
+        Sum(x*y, (x, 1, 3), (a, 2, 5)).doit().diff(y) == 24
+    assert Sum(x, (x, 1, 2)).diff(y) == 0
+
+
+def test_hypersum():
+    assert simplify(summation(x**n/fac(n), (n, 1, oo))) == -1 + exp(x)
+    assert summation((-1)**n * x**(2*n) / fac(2*n), (n, 0, oo)) == cos(x)
+    assert simplify(summation((-1)**n*x**(2*n + 1) /
+        factorial(2*n + 1), (n, 3, oo))) == -x + sin(x) + x**3/6 - x**5/120
+
+    assert summation(1/(n + 2)**3, (n, 1, oo)) == Rational(-9, 8) + zeta(3)
+    assert summation(1/n**4, (n, 1, oo)) == pi**4/90
+
+    s = summation(x**n*n, (n, -oo, 0))
+    assert s.is_Piecewise
+    assert s.args[0].args[0] == -1/(x*(1 - 1/x)**2)
+    assert s.args[0].args[1] == (abs(1/x) < 1)
+
+    m = Symbol('n', integer=True, positive=True)
+    assert summation(binomial(m, k), (k, 0, m)) == 2**m
+
+
+def test_issue_4170():
+    assert summation(1/factorial(k), (k, 0, oo)) == E
+
+
+def test_is_commutative():
+    from sympy.physics.secondquant import NO, F, Fd
+    m = Symbol('m', commutative=False)
+    for f in (Sum, Product, Integral):
+        assert f(z, (z, 1, 1)).is_commutative is True
+        assert f(z*y, (z, 1, 6)).is_commutative is True
+        assert f(m*x, (x, 1, 2)).is_commutative is False
+
+        assert f(NO(Fd(x)*F(y))*z, (z, 1, 2)).is_commutative is False
+
+
+def test_is_zero():
+    for func in [Sum, Product]:
+        assert func(0, (x, 1, 1)).is_zero is True
+        assert func(x, (x, 1, 1)).is_zero is None
+
+    assert Sum(0, (x, 1, 0)).is_zero is True
+    assert Product(0, (x, 1, 0)).is_zero is False
+
+
+def test_is_number():
+    # is number should not rely on evaluation or assumptions,
+    # it should be equivalent to `not foo.free_symbols`
+    assert Sum(1, (x, 1, 1)).is_number is True
+    assert Sum(1, (x, 1, x)).is_number is False
+    assert Sum(0, (x, y, z)).is_number is False
+    assert Sum(x, (y, 1, 2)).is_number is False
+    assert Sum(x, (y, 1, 1)).is_number is False
+    assert Sum(x, (x, 1, 2)).is_number is True
+    assert Sum(x*y, (x, 1, 2), (y, 1, 3)).is_number is True
+
+    assert Product(2, (x, 1, 1)).is_number is True
+    assert Product(2, (x, 1, y)).is_number is False
+    assert Product(0, (x, y, z)).is_number is False
+    assert Product(1, (x, y, z)).is_number is False
+    assert Product(x, (y, 1, x)).is_number is False
+    assert Product(x, (y, 1, 2)).is_number is False
+    assert Product(x, (y, 1, 1)).is_number is False
+    assert Product(x, (x, 1, 2)).is_number is True
+
+
+def test_free_symbols():
+    for func in [Sum, Product]:
+        assert func(1, (x, 1, 2)).free_symbols == set()
+        assert func(0, (x, 1, y)).free_symbols == {y}
+        assert func(2, (x, 1, y)).free_symbols == {y}
+        assert func(x, (x, 1, 2)).free_symbols == set()
+        assert func(x, (x, 1, y)).free_symbols == {y}
+        assert func(x, (y, 1, y)).free_symbols == {x, y}
+        assert func(x, (y, 1, 2)).free_symbols == {x}
+        assert func(x, (y, 1, 1)).free_symbols == {x}
+        assert func(x, (y, 1, z)).free_symbols == {x, z}
+        assert func(x, (x, 1, y), (y, 1, 2)).free_symbols == set()
+        assert func(x, (x, 1, y), (y, 1, z)).free_symbols == {z}
+        assert func(x, (x, 1, y), (y, 1, y)).free_symbols == {y}
+        assert func(x, (y, 1, y), (y, 1, z)).free_symbols == {x, z}
+    assert Sum(1, (x, 1, y)).free_symbols == {y}
+    # free_symbols answers whether the object *as written* has free symbols,
+    # not whether the evaluated expression has free symbols
+    assert Product(1, (x, 1, y)).free_symbols == {y}
+    # don't count free symbols that are not independent of integration
+    # variable(s)
+    assert func(f(x), (f(x), 1, 2)).free_symbols == set()
+    assert func(f(x), (f(x), 1, x)).free_symbols == {x}
+    assert func(f(x), (f(x), 1, y)).free_symbols == {y}
+    assert func(f(x), (z, 1, y)).free_symbols == {x, y}
+
+
+def test_conjugate_transpose():
+    A, B = symbols("A B", commutative=False)
+    p = Sum(A*B**n, (n, 1, 3))
+    assert p.adjoint().doit() == p.doit().adjoint()
+    assert p.conjugate().doit() == p.doit().conjugate()
+    assert p.transpose().doit() == p.doit().transpose()
+
+    p = Sum(B**n*A, (n, 1, 3))
+    assert p.adjoint().doit() == p.doit().adjoint()
+    assert p.conjugate().doit() == p.doit().conjugate()
+    assert p.transpose().doit() == p.doit().transpose()
+
+
+def test_noncommutativity_honoured():
+    A, B = symbols("A B", commutative=False)
+    M = symbols('M', integer=True, positive=True)
+    p = Sum(A*B**n, (n, 1, M))
+    assert p.doit() == A*Piecewise((M, Eq(B, 1)),
+                                   ((B - B**(M + 1))*(1 - B)**(-1), True))
+
+    p = Sum(B**n*A, (n, 1, M))
+    assert p.doit() == Piecewise((M, Eq(B, 1)),
+                                 ((B - B**(M + 1))*(1 - B)**(-1), True))*A
+
+    p = Sum(B**n*A*B**n, (n, 1, M))
+    assert p.doit() == p
+
+
+def test_issue_4171():
+    assert summation(factorial(2*k + 1)/factorial(2*k), (k, 0, oo)) is oo
+    assert summation(2*k + 1, (k, 0, oo)) is oo
+
+
+def test_issue_6273():
+    assert Sum(x, (x, 1, n)).n(2, subs={n: 1}) == Float(1, 2)
+
+
+def test_issue_6274():
+    assert Sum(x, (x, 1, 0)).doit() == 0
+    assert NS(Sum(x, (x, 1, 0))) == '0'
+    assert Sum(n, (n, 10, 5)).doit() == -30
+    assert NS(Sum(n, (n, 10, 5))) == '-30.0000000000000'
+
+
+def test_simplify_sum():
+    y, t, v = symbols('y, t, v')
+
+    _simplify = lambda e: simplify(e, doit=False)
+    assert _simplify(Sum(x*y, (x, n, m), (y, a, k)) + \
+        Sum(y, (x, n, m), (y, a, k))) == Sum(y * (x + 1), (x, n, m), (y, a, k))
+    assert _simplify(Sum(x, (x, n, m)) + Sum(x, (x, m + 1, a))) == \
+        Sum(x, (x, n, a))
+    assert _simplify(Sum(x, (x, k + 1, a)) + Sum(x, (x, n, k))) == \
+        Sum(x, (x, n, a))
+    assert _simplify(Sum(x, (x, k + 1, a)) + Sum(x + 1, (x, n, k))) == \
+        Sum(x, (x, n, a)) + Sum(1, (x, n, k))
+    assert _simplify(Sum(x, (x, 0, 3)) * 3 + 3 * Sum(x, (x, 4, 6)) + \
+        4 * Sum(z, (z, 0, 1))) == 4*Sum(z, (z, 0, 1)) + 3*Sum(x, (x, 0, 6))
+    assert _simplify(3*Sum(x**2, (x, a, b)) + Sum(x, (x, a, b))) == \
+        Sum(x*(3*x + 1), (x, a, b))
+    assert _simplify(Sum(x**3, (x, n, k)) * 3 + 3 * Sum(x, (x, n, k)) + \
+        4 * y * Sum(z, (z, n, k))) + 1 == \
+            4*y*Sum(z, (z, n, k)) + 3*Sum(x**3 + x, (x, n, k)) + 1
+    assert _simplify(Sum(x, (x, a, b)) + 1 + Sum(x, (x, b + 1, c))) == \
+        1 + Sum(x, (x, a, c))
+    assert _simplify(Sum(x, (t, a, b)) + Sum(y, (t, a, b)) + \
+        Sum(x, (t, b+1, c))) == x * Sum(1, (t, a, c)) + y * Sum(1, (t, a, b))
+    assert _simplify(Sum(x, (t, a, b)) + Sum(x, (t, b+1, c)) + \
+        Sum(y, (t, a, b))) == x * Sum(1, (t, a, c)) + y * Sum(1, (t, a, b))
+    assert _simplify(Sum(x, (t, a, b)) + 2 * Sum(x, (t, b+1, c))) == \
+        _simplify(Sum(x, (t, a, b)) + Sum(x, (t, b+1, c)) + Sum(x, (t, b+1, c)))
+    assert _simplify(Sum(x, (x, a, b))*Sum(x**2, (x, a, b))) == \
+        Sum(x, (x, a, b)) * Sum(x**2, (x, a, b))
+    assert _simplify(Sum(x, (t, a, b)) + Sum(y, (t, a, b)) + Sum(z, (t, a, b))) \
+        == (x + y + z) * Sum(1, (t, a, b))          # issue 8596
+    assert _simplify(Sum(x, (t, a, b)) + Sum(y, (t, a, b)) + Sum(z, (t, a, b)) + \
+        Sum(v, (t, a, b))) == (x + y + z + v) * Sum(1, (t, a, b))  # issue 8596
+    assert _simplify(Sum(x * y, (x, a, b)) / (3 * y)) == \
+        (Sum(x, (x, a, b)) / 3)
+    assert _simplify(Sum(f(x) * y * z, (x, a, b)) / (y * z)) \
+        == Sum(f(x), (x, a, b))
+    assert _simplify(Sum(c * x, (x, a, b)) - c * Sum(x, (x, a, b))) == 0
+    assert _simplify(c * (Sum(x, (x, a, b))  + y)) == c * (y + Sum(x, (x, a, b)))
+    assert _simplify(c * (Sum(x, (x, a, b)) + y * Sum(x, (x, a, b)))) == \
+        c * (y + 1) * Sum(x, (x, a, b))
+    assert _simplify(Sum(Sum(c * x, (x, a, b)), (y, a, b))) == \
+                c * Sum(x, (x, a, b), (y, a, b))
+    assert _simplify(Sum((3 + y) * Sum(c * x, (x, a, b)), (y, a, b))) == \
+                c * Sum((3 + y), (y, a, b)) * Sum(x, (x, a, b))
+    assert _simplify(Sum((3 + t) * Sum(c * t, (x, a, b)), (y, a, b))) == \
+                c*t*(t + 3)*Sum(1, (x, a, b))*Sum(1, (y, a, b))
+    assert _simplify(Sum(Sum(d * t, (x, a, b - 1)) + \
+                Sum(d * t, (x, b, c)), (t, a, b))) == \
+                    d * Sum(1, (x, a, c)) * Sum(t, (t, a, b))
+    assert _simplify(Sum(sin(t)**2 + cos(t)**2 + 1, (t, a, b))) == \
+        2 * Sum(1, (t, a, b))
+
+
+def test_change_index():
+    b, v, w = symbols('b, v, w', integer = True)
+
+    assert Sum(x, (x, a, b)).change_index(x, x + 1, y) == \
+        Sum(y - 1, (y, a + 1, b + 1))
+    assert Sum(x**2, (x, a, b)).change_index( x, x - 1) == \
+        Sum((x+1)**2, (x, a - 1, b - 1))
+    assert Sum(x**2, (x, a, b)).change_index( x, -x, y) == \
+        Sum((-y)**2, (y, -b, -a))
+    assert Sum(x, (x, a, b)).change_index( x, -x - 1) == \
+        Sum(-x - 1, (x, -b - 1, -a - 1))
+    assert Sum(x*y, (x, a, b), (y, c, d)).change_index( x, x - 1, z) == \
+        Sum((z + 1)*y, (z, a - 1, b - 1), (y, c, d))
+    assert Sum(x, (x, a, b)).change_index( x, x + v) == \
+        Sum(-v + x, (x, a + v, b + v))
+    assert Sum(x, (x, a, b)).change_index( x, -x - v) == \
+        Sum(-v - x, (x, -b - v, -a - v))
+    assert Sum(x, (x, a, b)).change_index(x, w*x, v) == \
+        Sum(v/w, (v, b*w, a*w))
+    raises(ValueError, lambda: Sum(x, (x, a, b)).change_index(x, 2*x))
+
+
+def test_reorder():
+    b, y, c, d, z = symbols('b, y, c, d, z', integer = True)
+
+    assert Sum(x*y, (x, a, b), (y, c, d)).reorder((0, 1)) == \
+        Sum(x*y, (y, c, d), (x, a, b))
+    assert Sum(x, (x, a, b), (x, c, d)).reorder((0, 1)) == \
+        Sum(x, (x, c, d), (x, a, b))
+    assert Sum(x*y + z, (x, a, b), (z, m, n), (y, c, d)).reorder(\
+        (2, 0), (0, 1)) == Sum(x*y + z, (z, m, n), (y, c, d), (x, a, b))
+    assert Sum(x*y*z, (x, a, b), (y, c, d), (z, m, n)).reorder(\
+        (0, 1), (1, 2), (0, 2)) == Sum(x*y*z, (x, a, b), (z, m, n), (y, c, d))
+    assert Sum(x*y*z, (x, a, b), (y, c, d), (z, m, n)).reorder(\
+        (x, y), (y, z), (x, z)) == Sum(x*y*z, (x, a, b), (z, m, n), (y, c, d))
+    assert Sum(x*y, (x, a, b), (y, c, d)).reorder((x, 1)) == \
+        Sum(x*y, (y, c, d), (x, a, b))
+    assert Sum(x*y, (x, a, b), (y, c, d)).reorder((y, x)) == \
+        Sum(x*y, (y, c, d), (x, a, b))
+
+
+def test_reverse_order():
+    assert Sum(x, (x, 0, 3)).reverse_order(0) == Sum(-x, (x, 4, -1))
+    assert Sum(x*y, (x, 1, 5), (y, 0, 6)).reverse_order(0, 1) == \
+           Sum(x*y, (x, 6, 0), (y, 7, -1))
+    assert Sum(x, (x, 1, 2)).reverse_order(0) == Sum(-x, (x, 3, 0))
+    assert Sum(x, (x, 1, 3)).reverse_order(0) == Sum(-x, (x, 4, 0))
+    assert Sum(x, (x, 1, a)).reverse_order(0) == Sum(-x, (x, a + 1, 0))
+    assert Sum(x, (x, a, 5)).reverse_order(0) == Sum(-x, (x, 6, a - 1))
+    assert Sum(x, (x, a + 1, a + 5)).reverse_order(0) == \
+                         Sum(-x, (x, a + 6, a))
+    assert Sum(x, (x, a + 1, a + 2)).reverse_order(0) == \
+           Sum(-x, (x, a + 3, a))
+    assert Sum(x, (x, a + 1, a + 1)).reverse_order(0) == \
+           Sum(-x, (x, a + 2, a))
+    assert Sum(x, (x, a, b)).reverse_order(0) == Sum(-x, (x, b + 1, a - 1))
+    assert Sum(x, (x, a, b)).reverse_order(x) == Sum(-x, (x, b + 1, a - 1))
+    assert Sum(x*y, (x, a, b), (y, 2, 5)).reverse_order(x, 1) == \
+        Sum(x*y, (x, b + 1, a - 1), (y, 6, 1))
+    assert Sum(x*y, (x, a, b), (y, 2, 5)).reverse_order(y, x) == \
+        Sum(x*y, (x, b + 1, a - 1), (y, 6, 1))
+
+
+def test_issue_7097():
+    assert sum(x**n/n for n in range(1, 401)) == summation(x**n/n, (n, 1, 400))
+
+
+def test_factor_expand_subs():
+    # test factoring
+    assert Sum(4 * x, (x, 1, y)).factor() == 4 * Sum(x, (x, 1, y))
+    assert Sum(x * a, (x, 1, y)).factor() == a * Sum(x, (x, 1, y))
+    assert Sum(4 * x * a, (x, 1, y)).factor() == 4 * a * Sum(x, (x, 1, y))
+    assert Sum(4 * x * y, (x, 1, y)).factor() == 4 * y * Sum(x, (x, 1, y))
+
+    # test expand
+    _x = Symbol('x', zero=False)
+    assert Sum(x+1,(x,1,y)).expand() == Sum(x,(x,1,y)) + Sum(1,(x,1,y))
+    assert Sum(x+a*x**2,(x,1,y)).expand() == Sum(x,(x,1,y)) + Sum(a*x**2,(x,1,y))
+    assert Sum(_x**(n + 1)*(n + 1), (n, -1, oo)).expand() \
+        == Sum(n*_x*_x**n + _x*_x**n, (n, -1, oo))
+    assert Sum(x**(n + 1)*(n + 1), (n, -1, oo)).expand(power_exp=False) \
+        == Sum(n*x**(n + 1) + x**(n + 1), (n, -1, oo))
+    assert Sum(x**(n + 1)*(n + 1), (n, -1, oo)).expand(force=True) \
+           == Sum(x*x**n, (n, -1, oo)) + Sum(n*x*x**n, (n, -1, oo))
+    assert Sum(a*n+a*n**2,(n,0,4)).expand() \
+        == Sum(a*n,(n,0,4)) + Sum(a*n**2,(n,0,4))
+    assert Sum(_x**a*_x**n,(x,0,3)) \
+        == Sum(_x**(a+n),(x,0,3)).expand(power_exp=True)
+    _a, _n = symbols('a n', positive=True)
+    assert Sum(x**(_a+_n),(x,0,3)).expand(power_exp=True) \
+        == Sum(x**_a*x**_n, (x, 0, 3))
+    assert Sum(x**(_a-_n),(x,0,3)).expand(power_exp=True) \
+        == Sum(x**(_a-_n),(x,0,3)).expand(power_exp=False)
+
+    # test subs
+    assert Sum(1/(1+a*x**2),(x,0,3)).subs([(a,3)]) == Sum(1/(1+3*x**2),(x,0,3))
+    assert Sum(x*y,(x,0,y),(y,0,x)).subs([(x,3)]) == Sum(x*y,(x,0,y),(y,0,3))
+    assert Sum(x,(x,1,10)).subs([(x,y-2)]) == Sum(x,(x,1,10))
+    assert Sum(1/x,(x,1,10)).subs([(x,(3+n)**3)]) == Sum(1/x,(x,1,10))
+    assert Sum(1/x,(x,1,10)).subs([(x,3*x-2)]) == Sum(1/x,(x,1,10))
+
+
+def test_distribution_over_equality():
+    assert Product(Eq(x*2, f(x)), (x, 1, 3)).doit() == Eq(48, f(1)*f(2)*f(3))
+    assert Sum(Eq(f(x), x**2), (x, 0, y)) == \
+        Eq(Sum(f(x), (x, 0, y)), Sum(x**2, (x, 0, y)))
+
+
+def test_issue_2787():
+    n, k = symbols('n k', positive=True, integer=True)
+    p = symbols('p', positive=True)
+    binomial_dist = binomial(n, k)*p**k*(1 - p)**(n - k)
+    s = Sum(binomial_dist*k, (k, 0, n))
+    res = s.doit().simplify()
+    ans = Piecewise(
+        (n*p, x),
+        (Sum(k*p**k*binomial(n, k)*(1 - p)**(n - k), (k, 0, n)),
+        True)).subs(x, (Eq(n, 1) | (n > 1)) & (p/Abs(p - 1) <= 1))
+    ans2 = Piecewise(
+        (n*p, x),
+        (factorial(n)*Sum(p**k*(1 - p)**(-k + n)/
+        (factorial(-k + n)*factorial(k - 1)), (k, 0, n)),
+        True)).subs(x, (Eq(n, 1) | (n > 1)) & (p/Abs(p - 1) <= 1))
+    assert res in [ans, ans2]  # XXX system dependent
+    # Issue #17165: make sure that another simplify does not complicate
+    # the result by much. Why didn't first simplify replace
+    # Eq(n, 1) | (n > 1) with True?
+    assert res.simplify().count_ops() <= res.count_ops() + 2
+
+
+def test_issue_4668():
+    assert summation(1/n, (n, 2, oo)) is oo
+
+
+def test_matrix_sum():
+    A = Matrix([[0, 1], [n, 0]])
+
+    result = Sum(A, (n, 0, 3)).doit()
+    assert result == Matrix([[0, 4], [6, 0]])
+    assert result.__class__ == ImmutableDenseMatrix
+
+    A = SparseMatrix([[0, 1], [n, 0]])
+
+    result = Sum(A, (n, 0, 3)).doit()
+    assert result.__class__ == ImmutableSparseMatrix
+
+
+def test_failing_matrix_sum():
+    n = Symbol('n')
+    # TODO Implement matrix geometric series summation.
+    A = Matrix([[0, 1, 0], [-1, 0, 0], [0, 0, 0]])
+    assert Sum(A ** n, (n, 1, 4)).doit() == \
+        Matrix([[0, 0, 0], [0, 0, 0], [0, 0, 0]])
+    # issue sympy/sympy#16989
+    assert summation(A**n, (n, 1, 1)) == A
+
+
+def test_indexed_idx_sum():
+    i = symbols('i', cls=Idx)
+    r = Indexed('r', i)
+    assert Sum(r, (i, 0, 3)).doit() == sum(r.xreplace({i: j}) for j in range(4))
+    assert Product(r, (i, 0, 3)).doit() == prod([r.xreplace({i: j}) for j in range(4)])
+
+    j = symbols('j', integer=True)
+    assert Sum(r, (i, j, j+2)).doit() == sum(r.xreplace({i: j+k}) for k in range(3))
+    assert Product(r, (i, j, j+2)).doit() == prod([r.xreplace({i: j+k}) for k in range(3)])
+
+    k = Idx('k', range=(1, 3))
+    A = IndexedBase('A')
+    assert Sum(A[k], k).doit() == sum(A[Idx(j, (1, 3))] for j in range(1, 4))
+    assert Product(A[k], k).doit() == prod([A[Idx(j, (1, 3))] for j in range(1, 4)])
+
+    raises(ValueError, lambda: Sum(A[k], (k, 1, 4)))
+    raises(ValueError, lambda: Sum(A[k], (k, 0, 3)))
+    raises(ValueError, lambda: Sum(A[k], (k, 2, oo)))
+
+    raises(ValueError, lambda: Product(A[k], (k, 1, 4)))
+    raises(ValueError, lambda: Product(A[k], (k, 0, 3)))
+    raises(ValueError, lambda: Product(A[k], (k, 2, oo)))
+
+
+@slow
+def test_is_convergent():
+    # divergence tests --
+    assert Sum(n/(2*n + 1), (n, 1, oo)).is_convergent() is S.false
+    assert Sum(factorial(n)/5**n, (n, 1, oo)).is_convergent() is S.false
+    assert Sum(3**(-2*n - 1)*n**n, (n, 1, oo)).is_convergent() is S.false
+    assert Sum((-1)**n*n, (n, 3, oo)).is_convergent() is S.false
+    assert Sum((-1)**n, (n, 1, oo)).is_convergent() is S.false
+    assert Sum(log(1/n), (n, 2, oo)).is_convergent() is S.false
+    assert Sum(sin(n), (n, 1, oo)).is_convergent() is S.false
+
+    # Raabe's test --
+    assert Sum(Product((3*m),(m,1,n))/Product((3*m+4),(m,1,n)),(n,1,oo)).is_convergent() is S.true
+
+    # root test --
+    assert Sum((-12)**n/n, (n, 1, oo)).is_convergent() is S.false
+
+    # integral test --
+
+    # p-series test --
+    assert Sum(1/(n**2 + 1), (n, 1, oo)).is_convergent() is S.true
+    assert Sum(1/n**Rational(6, 5), (n, 1, oo)).is_convergent() is S.true
+    assert Sum(2/(n*sqrt(n - 1)), (n, 2, oo)).is_convergent() is S.true
+    assert Sum(1/(sqrt(n)*sqrt(n)), (n, 2, oo)).is_convergent() is S.false
+    assert Sum(factorial(n) / factorial(n+2), (n, 1, oo)).is_convergent() is S.true
+    assert Sum(rf(5,n)/rf(7,n),(n,1,oo)).is_convergent() is S.true
+    assert Sum((rf(1, n)*rf(2, n))/(rf(3, n)*factorial(n)),(n,1,oo)).is_convergent() is S.false
+
+    # comparison test --
+    assert Sum(1/(n + log(n)), (n, 1, oo)).is_convergent() is S.false
+    assert Sum(1/(n**2*log(n)), (n, 2, oo)).is_convergent() is S.true
+    assert Sum(1/(n*log(n)), (n, 2, oo)).is_convergent() is S.false
+    assert Sum(2/(n*log(n)*log(log(n))**2), (n, 5, oo)).is_convergent() is S.true
+    assert Sum(2/(n*log(n)**2), (n, 2, oo)).is_convergent() is S.true
+    assert Sum((n - 1)/(n**2*log(n)**3), (n, 2, oo)).is_convergent() is S.true
+    assert Sum(1/(n*log(n)*log(log(n))), (n, 5, oo)).is_convergent() is S.false
+    assert Sum((n - 1)/(n*log(n)**3), (n, 3, oo)).is_convergent() is S.false
+    assert Sum(2/(n**2*log(n)), (n, 2, oo)).is_convergent() is S.true
+    assert Sum(1/(n*sqrt(log(n))*log(log(n))), (n, 100, oo)).is_convergent() is S.false
+    assert Sum(log(log(n))/(n*log(n)**2), (n, 100, oo)).is_convergent() is S.true
+    assert Sum(log(n)/n**2, (n, 5, oo)).is_convergent() is S.true
+
+    # alternating series tests --
+    assert Sum((-1)**(n - 1)/(n**2 - 1), (n, 3, oo)).is_convergent() is S.true
+
+    # with -negativeInfinite Limits
+    assert Sum(1/(n**2 + 1), (n, -oo, 1)).is_convergent() is S.true
+    assert Sum(1/(n - 1), (n, -oo, -1)).is_convergent() is S.false
+    assert Sum(1/(n**2 - 1), (n, -oo, -5)).is_convergent() is S.true
+    assert Sum(1/(n**2 - 1), (n, -oo, 2)).is_convergent() is S.true
+    assert Sum(1/(n**2 - 1), (n, -oo, oo)).is_convergent() is S.true
+
+    # piecewise functions
+    f = Piecewise((n**(-2), n <= 1), (n**2, n > 1))
+    assert Sum(f, (n, 1, oo)).is_convergent() is S.false
+    assert Sum(f, (n, -oo, oo)).is_convergent() is S.false
+    assert Sum(f, (n, 1, 100)).is_convergent() is S.true
+    #assert Sum(f, (n, -oo, 1)).is_convergent() is S.true
+
+    # integral test
+
+    assert Sum(log(n)/n**3, (n, 1, oo)).is_convergent() is S.true
+    assert Sum(-log(n)/n**3, (n, 1, oo)).is_convergent() is S.true
+    # the following function has maxima located at (x, y) =
+    # (1.2, 0.43), (3.0, -0.25) and (6.8, 0.050)
+    eq = (x - 2)*(x**2 - 6*x + 4)*exp(-x)
+    assert Sum(eq, (x, 1, oo)).is_convergent() is S.true
+    assert Sum(eq, (x, 1, 2)).is_convergent() is S.true
+    assert Sum(1/(x**3), (x, 1, oo)).is_convergent() is S.true
+    assert Sum(1/(x**S.Half), (x, 1, oo)).is_convergent() is S.false
+
+    # issue 19545
+    assert Sum(1/n - 3/(3*n +2), (n, 1, oo)).is_convergent() is S.true
+
+    # issue 19836
+    assert Sum(4/(n + 2) - 5/(n + 1) + 1/n,(n, 7, oo)).is_convergent() is S.true
+
+
+def test_is_absolutely_convergent():
+    assert Sum((-1)**n, (n, 1, oo)).is_absolutely_convergent() is S.false
+    assert Sum((-1)**n/n**2, (n, 1, oo)).is_absolutely_convergent() is S.true
+
+
+@XFAIL
+def test_convergent_failing():
+    # dirichlet tests
+    assert Sum(sin(n)/n, (n, 1, oo)).is_convergent() is S.true
+    assert Sum(sin(2*n)/n, (n, 1, oo)).is_convergent() is S.true
+
+
+def test_issue_6966():
+    i, k, m = symbols('i k m', integer=True)
+    z_i, q_i = symbols('z_i q_i')
+    a_k = Sum(-q_i*z_i/k,(i,1,m))
+    b_k = a_k.diff(z_i)
+    assert isinstance(b_k, Sum)
+    assert b_k == Sum(-q_i/k,(i,1,m))
+
+
+def test_issue_10156():
+    cx = Sum(2*y**2*x, (x, 1,3))
+    e = 2*y*Sum(2*cx*x**2, (x, 1, 9))
+    assert e.factor() == \
+        8*y**3*Sum(x, (x, 1, 3))*Sum(x**2, (x, 1, 9))
+
+
+def test_issue_10973():
+    assert Sum((-n + (n**3 + 1)**(S(1)/3))/log(n), (n, 1, oo)).is_convergent() is S.true
+
+
+def test_issue_14103():
+    assert Sum(sin(n)**2 + cos(n)**2 - 1, (n, 1, oo)).is_convergent() is S.true
+    assert Sum(sin(pi*n), (n, 1, oo)).is_convergent() is S.true
+
+
+def test_issue_14129():
+    x = Symbol('x', zero=False)
+    assert Sum( k*x**k, (k, 0, n-1)).doit() == \
+        Piecewise((n**2/2 - n/2, Eq(x, 1)), ((n*x*x**n -
+            n*x**n - x*x**n + x)/(x - 1)**2, True))
+    assert Sum( x**k, (k, 0, n-1)).doit() == \
+        Piecewise((n, Eq(x, 1)), ((-x**n + 1)/(-x + 1), True))
+    assert Sum( k*(x/y+x)**k, (k, 0, n-1)).doit() == \
+        Piecewise((n*(n - 1)/2, Eq(x, y/(y + 1))),
+        (x*(y + 1)*(n*x*y*(x + x/y)**(n - 1) +
+        n*x*(x + x/y)**(n - 1) - n*y*(x + x/y)**(n - 1) -
+        x*y*(x + x/y)**(n - 1) - x*(x + x/y)**(n - 1) + y)/
+        (x*y + x - y)**2, True))
+
+
+def test_issue_14112():
+    assert Sum((-1)**n/sqrt(n), (n, 1, oo)).is_absolutely_convergent() is S.false
+    assert Sum((-1)**(2*n)/n, (n, 1, oo)).is_convergent() is S.false
+    assert Sum((-2)**n + (-3)**n, (n, 1, oo)).is_convergent() is S.false
+
+
+def test_issue_14219():
+    A = diag(0, 2, -3)
+    res = diag(1, 15, -20)
+    assert Sum(A**n, (n, 0, 3)).doit() == res
+
+
+def test_sin_times_absolutely_convergent():
+    assert Sum(sin(n) / n**3, (n, 1, oo)).is_convergent() is S.true
+    assert Sum(sin(n) * log(n) / n**3, (n, 1, oo)).is_convergent() is S.true
+
+
+def test_issue_14111():
+    assert Sum(1/log(log(n)), (n, 22, oo)).is_convergent() is S.false
+
+
+def test_issue_14484():
+    assert Sum(sin(n)/log(log(n)), (n, 22, oo)).is_convergent() is S.false
+
+
+def test_issue_14640():
+    i, n = symbols("i n", integer=True)
+    a, b, c = symbols("a b c", zero=False)
+
+    assert Sum(a**-i/(a - b), (i, 0, n)).doit() == Sum(
+        1/(a*a**i - a**i*b), (i, 0, n)).doit() == Piecewise(
+            (n + 1, Eq(1/a, 1)),
+            ((-a**(-n - 1) + 1)/(1 - 1/a), True))/(a - b)
+
+    assert Sum((b*a**i - c*a**i)**-2, (i, 0, n)).doit() == Piecewise(
+        (n + 1, Eq(a**(-2), 1)),
+        ((-a**(-2*n - 2) + 1)/(1 - 1/a**2), True))/(b - c)**2
+
+    s = Sum(i*(a**(n - i) - b**(n - i))/(a - b), (i, 0, n)).doit()
+    assert not s.has(Sum)
+    assert s.subs({a: 2, b: 3, n: 5}) == 122
+
+
+def test_issue_15943():
+    s = Sum(binomial(n, k)*factorial(n - k), (k, 0, n)).doit().rewrite(gamma)
+    assert s == -E*(n + 1)*gamma(n + 1)*lowergamma(n + 1, 1)/gamma(n + 2
+        ) + E*gamma(n + 1)
+    assert s.simplify() == E*(factorial(n) - lowergamma(n + 1, 1))
+
+
+def test_Sum_dummy_eq():
+    assert not Sum(x, (x, a, b)).dummy_eq(1)
+    assert not Sum(x, (x, a, b)).dummy_eq(Sum(x, (x, a, b), (a, 1, 2)))
+    assert not Sum(x, (x, a, b)).dummy_eq(Sum(x, (x, a, c)))
+    assert Sum(x, (x, a, b)).dummy_eq(Sum(x, (x, a, b)))
+    d = Dummy()
+    assert Sum(x, (x, a, d)).dummy_eq(Sum(x, (x, a, c)), c)
+    assert not Sum(x, (x, a, d)).dummy_eq(Sum(x, (x, a, c)))
+    assert Sum(x, (x, a, c)).dummy_eq(Sum(y, (y, a, c)))
+    assert Sum(x, (x, a, d)).dummy_eq(Sum(y, (y, a, c)), c)
+    assert not Sum(x, (x, a, d)).dummy_eq(Sum(y, (y, a, c)))
+
+
+def test_issue_15852():
+    assert summation(x**y*y, (y, -oo, oo)).doit() == Sum(x**y*y, (y, -oo, oo))
+
+
+def test_exceptions():
+    S = Sum(x, (x, a, b))
+    raises(ValueError, lambda: S.change_index(x, x**2, y))
+    S = Sum(x, (x, a, b), (x, 1, 4))
+    raises(ValueError, lambda: S.index(x))
+    S = Sum(x, (x, a, b), (y, 1, 4))
+    raises(ValueError, lambda: S.reorder([x]))
+    S = Sum(x, (x, y, b), (y, 1, 4))
+    raises(ReorderError, lambda: S.reorder_limit(0, 1))
+    S = Sum(x*y, (x, a, b), (y, 1, 4))
+    raises(NotImplementedError, lambda: S.is_convergent())
+
+
+def test_sumproducts_assumptions():
+    M = Symbol('M', integer=True, positive=True)
+
+    m = Symbol('m', integer=True)
+    for func in [Sum, Product]:
+        assert func(m, (m, -M, M)).is_positive is None
+        assert func(m, (m, -M, M)).is_nonpositive is None
+        assert func(m, (m, -M, M)).is_negative is None
+        assert func(m, (m, -M, M)).is_nonnegative is None
+        assert func(m, (m, -M, M)).is_finite is True
+
+    m = Symbol('m', integer=True, nonnegative=True)
+    for func in [Sum, Product]:
+        assert func(m, (m, 0, M)).is_positive is None
+        assert func(m, (m, 0, M)).is_nonpositive is None
+        assert func(m, (m, 0, M)).is_negative is False
+        assert func(m, (m, 0, M)).is_nonnegative is True
+        assert func(m, (m, 0, M)).is_finite is True
+
+    m = Symbol('m', integer=True, positive=True)
+    for func in [Sum, Product]:
+        assert func(m, (m, 1, M)).is_positive is True
+        assert func(m, (m, 1, M)).is_nonpositive is False
+        assert func(m, (m, 1, M)).is_negative is False
+        assert func(m, (m, 1, M)).is_nonnegative is True
+        assert func(m, (m, 1, M)).is_finite is True
+
+    m = Symbol('m', integer=True, negative=True)
+    assert Sum(m, (m, -M, -1)).is_positive is False
+    assert Sum(m, (m, -M, -1)).is_nonpositive is True
+    assert Sum(m, (m, -M, -1)).is_negative is True
+    assert Sum(m, (m, -M, -1)).is_nonnegative is False
+    assert Sum(m, (m, -M, -1)).is_finite is True
+    assert Product(m, (m, -M, -1)).is_positive is None
+    assert Product(m, (m, -M, -1)).is_nonpositive is None
+    assert Product(m, (m, -M, -1)).is_negative is None
+    assert Product(m, (m, -M, -1)).is_nonnegative is None
+    assert Product(m, (m, -M, -1)).is_finite is True
+
+    m = Symbol('m', integer=True, nonpositive=True)
+    assert Sum(m, (m, -M, 0)).is_positive is False
+    assert Sum(m, (m, -M, 0)).is_nonpositive is True
+    assert Sum(m, (m, -M, 0)).is_negative is None
+    assert Sum(m, (m, -M, 0)).is_nonnegative is None
+    assert Sum(m, (m, -M, 0)).is_finite is True
+    assert Product(m, (m, -M, 0)).is_positive is None
+    assert Product(m, (m, -M, 0)).is_nonpositive is None
+    assert Product(m, (m, -M, 0)).is_negative is None
+    assert Product(m, (m, -M, 0)).is_nonnegative is None
+    assert Product(m, (m, -M, 0)).is_finite is True
+
+    m = Symbol('m', integer=True)
+    assert Sum(2, (m, 0, oo)).is_positive is None
+    assert Sum(2, (m, 0, oo)).is_nonpositive is None
+    assert Sum(2, (m, 0, oo)).is_negative is None
+    assert Sum(2, (m, 0, oo)).is_nonnegative is None
+    assert Sum(2, (m, 0, oo)).is_finite is None
+
+    assert Product(2, (m, 0, oo)).is_positive is None
+    assert Product(2, (m, 0, oo)).is_nonpositive is None
+    assert Product(2, (m, 0, oo)).is_negative is False
+    assert Product(2, (m, 0, oo)).is_nonnegative is None
+    assert Product(2, (m, 0, oo)).is_finite is None
+
+    assert Product(0, (x, M, M-1)).is_positive is True
+    assert Product(0, (x, M, M-1)).is_finite is True
+
+
+def test_expand_with_assumptions():
+    M = Symbol('M', integer=True, positive=True)
+    x = Symbol('x', positive=True)
+    m = Symbol('m', nonnegative=True)
+    assert log(Product(x**m, (m, 0, M))).expand() == Sum(m*log(x), (m, 0, M))
+    assert log(Product(exp(x**m), (m, 0, M))).expand() == Sum(x**m, (m, 0, M))
+    assert log(Product(x**m, (m, 0, M))).rewrite(Sum).expand() == Sum(m*log(x), (m, 0, M))
+    assert log(Product(exp(x**m), (m, 0, M))).rewrite(Sum).expand() == Sum(x**m, (m, 0, M))
+
+    n = Symbol('n', nonnegative=True)
+    i, j = symbols('i,j', positive=True, integer=True)
+    x, y = symbols('x,y', positive=True)
+    assert log(Product(x**i*y**j, (i, 1, n), (j, 1, m))).expand() \
+        == Sum(i*log(x) + j*log(y), (i, 1, n), (j, 1, m))
+
+    m = Symbol('m', nonnegative=True, integer=True)
+    s = Sum(x**m, (m, 0, M))
+    s_as_product = s.rewrite(Product)
+    assert s_as_product.has(Product)
+    assert s_as_product == log(Product(exp(x**m), (m, 0, M)))
+    assert s_as_product.expand() == s
+    s5 = s.subs(M, 5)
+    s5_as_product = s5.rewrite(Product)
+    assert s5_as_product.has(Product)
+    assert s5_as_product.doit().expand() == s5.doit()
+
+
+def test_has_finite_limits():
+    x = Symbol('x')
+    assert Sum(1, (x, 1, 9)).has_finite_limits is True
+    assert Sum(1, (x, 1, oo)).has_finite_limits is False
+    M = Symbol('M')
+    assert Sum(1, (x, 1, M)).has_finite_limits is None
+    M = Symbol('M', positive=True)
+    assert Sum(1, (x, 1, M)).has_finite_limits is True
+    x = Symbol('x', positive=True)
+    M = Symbol('M')
+    assert Sum(1, (x, 1, M)).has_finite_limits is True
+
+    assert Sum(1, (x, 1, M), (y, -oo, oo)).has_finite_limits is False
+
+def test_has_reversed_limits():
+    assert Sum(1, (x, 1, 1)).has_reversed_limits is False
+    assert Sum(1, (x, 1, 9)).has_reversed_limits is False
+    assert Sum(1, (x, 1, -9)).has_reversed_limits is True
+    assert Sum(1, (x, 1, 0)).has_reversed_limits is True
+    assert Sum(1, (x, 1, oo)).has_reversed_limits is False
+    M = Symbol('M')
+    assert Sum(1, (x, 1, M)).has_reversed_limits is None
+    M = Symbol('M', positive=True, integer=True)
+    assert Sum(1, (x, 1, M)).has_reversed_limits is False
+    assert Sum(1, (x, 1, M), (y, -oo, oo)).has_reversed_limits is False
+    M = Symbol('M', negative=True)
+    assert Sum(1, (x, 1, M)).has_reversed_limits is True
+
+    assert Sum(1, (x, 1, M), (y, -oo, oo)).has_reversed_limits is True
+    assert Sum(1, (x, oo, oo)).has_reversed_limits is None
+
+
+def test_has_empty_sequence():
+    assert Sum(1, (x, 1, 1)).has_empty_sequence is False
+    assert Sum(1, (x, 1, 9)).has_empty_sequence is False
+    assert Sum(1, (x, 1, -9)).has_empty_sequence is False
+    assert Sum(1, (x, 1, 0)).has_empty_sequence is True
+    assert Sum(1, (x, y, y - 1)).has_empty_sequence is True
+    assert Sum(1, (x, 3, 2), (y, -oo, oo)).has_empty_sequence is True
+    assert Sum(1, (y, -oo, oo), (x, 3, 2)).has_empty_sequence is True
+    assert Sum(1, (x, oo, oo)).has_empty_sequence is False
+
+
+def test_empty_sequence():
+    assert Product(x*y, (x, -oo, oo), (y, 1, 0)).doit() == 1
+    assert Product(x*y, (y, 1, 0), (x, -oo, oo)).doit() == 1
+    assert Sum(x, (x, -oo, oo), (y, 1, 0)).doit() == 0
+    assert Sum(x, (y, 1, 0), (x, -oo, oo)).doit() == 0
+
+
+def test_issue_8016():
+    k = Symbol('k', integer=True)
+    n, m = symbols('n, m', integer=True, positive=True)
+    s = Sum(binomial(m, k)*binomial(m, n - k)*(-1)**k, (k, 0, n))
+    assert s.doit().simplify() == \
+        cos(pi*n/2)*gamma(m + 1)/gamma(n/2 + 1)/gamma(m - n/2 + 1)
+
+
+def test_issue_14313():
+    assert Sum(S.Half**floor(n/2), (n, 1, oo)).is_convergent()
+
+
+def test_issue_14563():
+    # The assertion was failing due to no assumptions methods in Sums and Product
+    assert 1 % Sum(1, (x, 0, 1)) == 1
+
+
+def test_issue_16735():
+    assert Sum(5**n/gamma(n+1), (n, 1, oo)).is_convergent() is S.true
+
+
+def test_issue_14871():
+    assert Sum((Rational(1, 10))**n*rf(0, n)/factorial(n), (n, 0, oo)).rewrite(factorial).doit() == 1
+
+
+def test_issue_17165():
+    n = symbols("n", integer=True)
+    x = symbols('x')
+    s = (x*Sum(x**n, (n, -1, oo)))
+    ssimp = s.doit().simplify()
+
+    assert ssimp == Piecewise((-1/(x - 1), (x > -1) & (x < 1)),
+                              (x*Sum(x**n, (n, -1, oo)), True)), ssimp
+    assert ssimp.simplify() == ssimp
+
+
+def test_issue_19379():
+    assert Sum(factorial(n)/factorial(n + 2), (n, 1, oo)).is_convergent() is S.true
+
+
+def test_issue_20777():
+    assert Sum(exp(x*sin(n/m)), (n, 1, m)).doit() == Sum(exp(x*sin(n/m)), (n, 1, m))
+
+
+def test__dummy_with_inherited_properties_concrete():
+    x = Symbol('x')
+
+    from sympy.core.containers import Tuple
+    d = _dummy_with_inherited_properties_concrete(Tuple(x, 0, 5))
+    assert d.is_real
+    assert d.is_integer
+    assert d.is_nonnegative
+    assert d.is_extended_nonnegative
+
+    d = _dummy_with_inherited_properties_concrete(Tuple(x, 1, 9))
+    assert d.is_real
+    assert d.is_integer
+    assert d.is_positive
+    assert d.is_odd is None
+
+    d = _dummy_with_inherited_properties_concrete(Tuple(x, -5, 5))
+    assert d.is_real
+    assert d.is_integer
+    assert d.is_positive is None
+    assert d.is_extended_nonnegative is None
+    assert d.is_odd is None
+
+    d = _dummy_with_inherited_properties_concrete(Tuple(x, -1.5, 1.5))
+    assert d.is_real
+    assert d.is_integer is None
+    assert d.is_positive is None
+    assert d.is_extended_nonnegative is None
+
+    N = Symbol('N', integer=True, positive=True)
+    d = _dummy_with_inherited_properties_concrete(Tuple(x, 2, N))
+    assert d.is_real
+    assert d.is_positive
+    assert d.is_integer
+
+    # Return None if no assumptions are added
+    N = Symbol('N', integer=True, positive=True)
+    d = _dummy_with_inherited_properties_concrete(Tuple(N, 2, 4))
+    assert d is None
+
+    x = Symbol('x', negative=True)
+    raises(InconsistentAssumptions,
+           lambda: _dummy_with_inherited_properties_concrete(Tuple(x, 1, 5)))
+
+
+def test_matrixsymbol_summation_numerical_limits():
+    A = MatrixSymbol('A', 3, 3)
+    n = Symbol('n', integer=True)
+
+    assert Sum(A**n, (n, 0, 2)).doit() == Identity(3) + A + A**2
+    assert Sum(A, (n, 0, 2)).doit() == 3*A
+    assert Sum(n*A, (n, 0, 2)).doit() == 3*A
+
+    B = Matrix([[0, n, 0], [-1, 0, 0], [0, 0, 2]])
+    ans = Matrix([[0, 6, 0], [-4, 0, 0], [0, 0, 8]]) + 4*A
+    assert Sum(A+B, (n, 0, 3)).doit() == ans
+    ans = A*Matrix([[0, 6, 0], [-4, 0, 0], [0, 0, 8]])
+    assert Sum(A*B, (n, 0, 3)).doit() == ans
+
+    ans = (A**2*Matrix([[-2, 0, 0], [0,-2, 0], [0, 0, 4]]) +
+           A**3*Matrix([[0, -9, 0], [3, 0, 0], [0, 0, 8]]) +
+           A*Matrix([[0, 1, 0], [-1, 0, 0], [0, 0, 2]]))
+    assert Sum(A**n*B**n, (n, 1, 3)).doit() == ans
+
+
+def test_issue_21651():
+    i = Symbol('i')
+    a = Sum(floor(2*2**(-i)), (i, S.One, 2))
+    assert a.doit() == S.One
+
+
+@XFAIL
+def test_matrixsymbol_summation_symbolic_limits():
+    N = Symbol('N', integer=True, positive=True)
+
+    A = MatrixSymbol('A', 3, 3)
+    n = Symbol('n', integer=True)
+    assert Sum(A, (n, 0, N)).doit() == (N+1)*A
+    assert Sum(n*A, (n, 0, N)).doit() == (N**2/2+N/2)*A
+
+
+def test_summation_by_residues():
+    x = Symbol('x')
+
+    # Examples from Nakhle H. Asmar, Loukas Grafakos,
+    # Complex Analysis with Applications
+    assert eval_sum_residue(1 / (x**2 + 1), (x, -oo, oo)) == pi/tanh(pi)
+    assert eval_sum_residue(1 / x**6, (x, S(1), oo)) == pi**6/945
+    assert eval_sum_residue(1 / (x**2 + 9), (x, -oo, oo)) == pi/(3*tanh(3*pi))
+    assert eval_sum_residue(1 / (x**2 + 1)**2, (x, -oo, oo)).cancel() == \
+        (-pi**2*tanh(pi)**2 + pi*tanh(pi) + pi**2)/(2*tanh(pi)**2)
+    assert eval_sum_residue(x**2 / (x**2 + 1)**2, (x, -oo, oo)).cancel() == \
+        (-pi**2 + pi*tanh(pi) + pi**2*tanh(pi)**2)/(2*tanh(pi)**2)
+    assert eval_sum_residue(1 / (4*x**2 - 1), (x, -oo, oo)) == 0
+    assert eval_sum_residue(x**2 / (x**2 - S(1)/4)**2, (x, -oo, oo)) == pi**2/2
+    assert eval_sum_residue(1 / (4*x**2 - 1)**2, (x, -oo, oo)) == pi**2/8
+    assert eval_sum_residue(1 / ((x - S(1)/2)**2 + 1), (x, -oo, oo)) == pi*tanh(pi)
+    assert eval_sum_residue(1 / x**2, (x, S(1), oo)) == pi**2/6
+    assert eval_sum_residue(1 / x**4, (x, S(1), oo)) == pi**4/90
+    assert eval_sum_residue(1 / x**2 / (x**2 + 4), (x, S(1), oo)) == \
+        -pi*(-pi/12 - 1/(16*pi) + 1/(8*tanh(2*pi)))/2
+
+    # Some examples made from 1 / (x**2 + 1)
+    assert eval_sum_residue(1 / (x**2 + 1), (x, S(0), oo)) == \
+        S(1)/2 + pi/(2*tanh(pi))
+    assert eval_sum_residue(1 / (x**2 + 1), (x, S(1), oo)) == \
+        -S(1)/2 + pi/(2*tanh(pi))
+    assert eval_sum_residue(1 / (x**2 + 1), (x, S(-1), oo)) == \
+        1 + pi/(2*tanh(pi))
+    assert eval_sum_residue((-1)**x / (x**2 + 1), (x, -oo, oo)) == \
+        pi/sinh(pi)
+    assert eval_sum_residue((-1)**x / (x**2 + 1), (x, S(0), oo)) == \
+        pi/(2*sinh(pi)) + S(1)/2
+    assert eval_sum_residue((-1)**x / (x**2 + 1), (x, S(1), oo)) == \
+        -S(1)/2 + pi/(2*sinh(pi))
+    assert eval_sum_residue((-1)**x / (x**2 + 1), (x, S(-1), oo)) == \
+        pi/(2*sinh(pi))
+
+    # Some examples made from shifting of 1 / (x**2 + 1)
+    assert eval_sum_residue(1 / (x**2 + 2*x + 2), (x, S(-1), oo)) == S(1)/2 + pi/(2*tanh(pi))
+    assert eval_sum_residue(1 / (x**2 + 4*x + 5), (x, S(-2), oo)) == S(1)/2 + pi/(2*tanh(pi))
+    assert eval_sum_residue(1 / (x**2 - 2*x + 2), (x, S(1), oo)) == S(1)/2 + pi/(2*tanh(pi))
+    assert eval_sum_residue(1 / (x**2 - 4*x + 5), (x, S(2), oo)) == S(1)/2 + pi/(2*tanh(pi))
+    assert eval_sum_residue((-1)**x * -1 / (x**2 + 2*x + 2), (x, S(-1), oo)) ==  S(1)/2 + pi/(2*sinh(pi))
+    assert eval_sum_residue((-1)**x * -1 / (x**2 -2*x + 2), (x, S(1), oo)) == S(1)/2 + pi/(2*sinh(pi))
+
+    # Some examples made from 1 / x**2
+    assert eval_sum_residue(1 / x**2, (x, S(2), oo)) == -1 + pi**2/6
+    assert eval_sum_residue(1 / x**2, (x, S(3), oo)) == -S(5)/4 + pi**2/6
+    assert eval_sum_residue((-1)**x / x**2, (x, S(1), oo)) == -pi**2/12
+    assert eval_sum_residue((-1)**x / x**2, (x, S(2), oo)) == 1 - pi**2/12
+
+
+@slow
+def test_summation_by_residues_failing():
+    x = Symbol('x')
+
+    # Failing because of the bug in residue computation
+    assert eval_sum_residue(x**2 / (x**4 + 1), (x, S(1), oo))
+    assert eval_sum_residue(1 / ((x - 1)*(x - 2) + 1), (x, -oo, oo)) != 0
+
+
+def test_process_limits():
+    from sympy.concrete.expr_with_limits import _process_limits
+
+    # these should be (x, Range(3)) not Range(3)
+    raises(ValueError, lambda: _process_limits(
+        Range(3), discrete=True))
+    raises(ValueError, lambda: _process_limits(
+        Range(3), discrete=False))
+    # these should be (x, union) not union
+    # (but then we would get a TypeError because we don't
+    # handle non-contiguous sets: see below use of `union`)
+    union = Or(x < 1, x > 3).as_set()
+    raises(ValueError, lambda: _process_limits(
+        union, discrete=True))
+    raises(ValueError, lambda: _process_limits(
+        union, discrete=False))
+
+    # error not triggered if not needed
+    assert _process_limits((x, 1, 2)) == ([(x, 1, 2)], 1)
+
+    # this equivalence is used to detect Reals in _process_limits
+    assert isinstance(S.Reals, Interval)
+
+    C = Integral  # continuous limits
+    assert C(x, x >= 5) == C(x, (x, 5, oo))
+    assert C(x, x < 3) == C(x, (x, -oo, 3))
+    ans = C(x, (x, 0, 3))
+    assert C(x, And(x >= 0, x < 3)) == ans
+    assert C(x, (x, Interval.Ropen(0, 3))) == ans
+    raises(TypeError, lambda: C(x, (x, Range(3))))
+
+    # discrete limits
+    for D in (Sum, Product):
+        r, ans = Range(3, 10, 2), D(2*x + 3, (x, 0, 3))
+        assert D(x, (x, r)) == ans
+        assert D(x, (x, r.reversed)) == ans
+        r, ans = Range(3, oo, 2), D(2*x + 3, (x, 0, oo))
+        assert D(x, (x, r)) == ans
+        assert D(x, (x, r.reversed)) == ans
+        r, ans = Range(-oo, 5, 2), D(3 - 2*x, (x, 0, oo))
+        assert D(x, (x, r)) == ans
+        assert D(x, (x, r.reversed)) == ans
+        raises(TypeError, lambda: D(x, x > 0))
+        raises(ValueError, lambda: D(x, Interval(1, 3)))
+        raises(NotImplementedError, lambda: D(x, (x, union)))
+
+
+def test_pr_22677():
+    b = Symbol('b', integer=True, positive=True)
+    assert Sum(1/x**2,(x, 0, b)).doit() == Sum(x**(-2), (x, 0, b))
+    assert Sum(1/(x - b)**2,(x, 0, b-1)).doit() == Sum(
+        (-b + x)**(-2), (x, 0, b - 1))
+
+
+def test_issue_23952():
+    p, q = symbols("p q", real=True, nonnegative=True)
+    k1, k2 = symbols("k1 k2", integer=True, nonnegative=True)
+    n = Symbol("n", integer=True, positive=True)
+    expr = Sum(abs(k1 - k2)*p**k1 *(1 - q)**(n - k2),
+        (k1, 0, n), (k2, 0, n))
+    assert expr.subs(p,0).subs(q,1).subs(n, 3).doit() == 3

.venv/lib/python3.13/site-packages/sympy/holonomic/tests/test_holonomic.py

@@ -0,0 +1,851 @@
+from sympy.holonomic import (DifferentialOperator, HolonomicFunction,
+                             DifferentialOperators, from_hyper,
+                             from_meijerg, expr_to_holonomic)
+from sympy.holonomic.recurrence import RecurrenceOperators, HolonomicSequence
+from sympy.core import EulerGamma
+from sympy.core.numbers import (I, Rational, pi)
+from sympy.core.singleton import S
+from sympy.core.symbol import (Symbol, symbols)
+from sympy.functions.elementary.exponential import (exp, log)
+from sympy.functions.elementary.hyperbolic import (asinh, cosh)
+from sympy.functions.elementary.miscellaneous import sqrt
+from sympy.functions.elementary.trigonometric import (cos, sin)
+from sympy.functions.special.bessel import besselj
+from sympy.functions.special.beta_functions import beta
+from sympy.functions.special.error_functions import (Ci, Si, erf, erfc)
+from sympy.functions.special.gamma_functions import gamma
+from sympy.functions.special.hyper import (hyper, meijerg)
+from sympy.printing.str import sstr
+from sympy.series.order import O
+from sympy.simplify.hyperexpand import hyperexpand
+from sympy.polys.domains.integerring import ZZ
+from sympy.polys.domains.rationalfield import QQ
+from sympy.polys.domains.realfield import RR
+
+
+def test_DifferentialOperator():
+    x = symbols('x')
+    R, Dx = DifferentialOperators(QQ.old_poly_ring(x), 'Dx')
+    assert Dx == R.derivative_operator
+    assert Dx == DifferentialOperator([R.base.zero, R.base.one], R)
+    assert x * Dx + x**2 * Dx**2 == DifferentialOperator([0, x, x**2], R)
+    assert (x**2 + 1) + Dx + x * \
+        Dx**5 == DifferentialOperator([x**2 + 1, 1, 0, 0, 0, x], R)
+    assert (x * Dx + x**2 + 1 - Dx * (x**3 + x))**3 == (-48 * x**6) + \
+        (-57 * x**7) * Dx + (-15 * x**8) * Dx**2 + (-x**9) * Dx**3
+    p = (x * Dx**2 + (x**2 + 3) * Dx**5) * (Dx + x**2)
+    q = (2 * x) + (4 * x**2) * Dx + (x**3) * Dx**2 + \
+        (20 * x**2 + x + 60) * Dx**3 + (10 * x**3 + 30 * x) * Dx**4 + \
+        (x**4 + 3 * x**2) * Dx**5 + (x**2 + 3) * Dx**6
+    assert p == q
+
+
+def test_HolonomicFunction_addition():
+    x = symbols('x')
+    R, Dx = DifferentialOperators(ZZ.old_poly_ring(x), 'Dx')
+    p = HolonomicFunction(Dx**2 * x, x)
+    q = HolonomicFunction((2) * Dx + (x) * Dx**2, x)
+    assert p == q
+    p = HolonomicFunction(x * Dx + 1, x)
+    q = HolonomicFunction(Dx + 1, x)
+    r = HolonomicFunction((x - 2) + (x**2 - 2) * Dx + (x**2 - x) * Dx**2, x)
+    assert p + q == r
+    p = HolonomicFunction(x * Dx + Dx**2 * (x**2 + 2), x)
+    q = HolonomicFunction(Dx - 3, x)
+    r = HolonomicFunction((-54 * x**2 - 126 * x - 150) + (-135 * x**3 - 252 * x**2 - 270 * x + 140) * Dx +\
+                 (-27 * x**4 - 24 * x**2 + 14 * x - 150) * Dx**2 + \
+                 (9 * x**4 + 15 * x**3 + 38 * x**2 + 30 * x +40) * Dx**3, x)
+    assert p + q == r
+    p = HolonomicFunction(Dx**5 - 1, x)
+    q = HolonomicFunction(x**3 + Dx, x)
+    r = HolonomicFunction((-x**18 + 45*x**14 - 525*x**10 + 1575*x**6 - x**3 - 630*x**2) + \
+        (-x**15 + 30*x**11 - 195*x**7 + 210*x**3 - 1)*Dx + (x**18 - 45*x**14 + 525*x**10 - \
+        1575*x**6 + x**3 + 630*x**2)*Dx**5 + (x**15 - 30*x**11 + 195*x**7 - 210*x**3 + \
+        1)*Dx**6, x)
+    assert p+q == r
+
+    p = x**2 + 3*x + 8
+    q = x**3 - 7*x + 5
+    p = p*Dx - p.diff()
+    q = q*Dx - q.diff()
+    r = HolonomicFunction(p, x) + HolonomicFunction(q, x)
+    s = HolonomicFunction((6*x**2 + 18*x + 14) + (-4*x**3 - 18*x**2 - 62*x + 10)*Dx +\
+        (x**4 + 6*x**3 + 31*x**2 - 10*x - 71)*Dx**2, x)
+    assert r == s
+
+
+def test_HolonomicFunction_multiplication():
+    x = symbols('x')
+    R, Dx = DifferentialOperators(ZZ.old_poly_ring(x), 'Dx')
+    p = HolonomicFunction(Dx+x+x*Dx**2, x)
+    q = HolonomicFunction(x*Dx+Dx*x+Dx**2, x)
+    r = HolonomicFunction((8*x**6 + 4*x**4 + 6*x**2 + 3) + (24*x**5 - 4*x**3 + 24*x)*Dx + \
+        (8*x**6 + 20*x**4 + 12*x**2 + 2)*Dx**2 + (8*x**5 + 4*x**3 + 4*x)*Dx**3 + \
+        (2*x**4 + x**2)*Dx**4, x)
+    assert p*q == r
+    p = HolonomicFunction(Dx**2+1, x)
+    q = HolonomicFunction(Dx-1, x)
+    r = HolonomicFunction((2) + (-2)*Dx + (1)*Dx**2, x)
+    assert p*q == r
+    p = HolonomicFunction(Dx**2+1+x+Dx, x)
+    q = HolonomicFunction((Dx*x-1)**2, x)
+    r = HolonomicFunction((4*x**7 + 11*x**6 + 16*x**5 + 4*x**4 - 6*x**3 - 7*x**2 - 8*x - 2) + \
+        (8*x**6 + 26*x**5 + 24*x**4 - 3*x**3 - 11*x**2 - 6*x - 2)*Dx + \
+        (8*x**6 + 18*x**5 + 15*x**4 - 3*x**3 - 6*x**2 - 6*x - 2)*Dx**2 + (8*x**5 + \
+            10*x**4 + 6*x**3 - 2*x**2 - 4*x)*Dx**3 + (4*x**5 + 3*x**4 - x**2)*Dx**4, x)
+    assert p*q == r
+    p = HolonomicFunction(x*Dx**2-1, x)
+    q = HolonomicFunction(Dx*x-x, x)
+    r = HolonomicFunction((x - 3) + (-2*x + 2)*Dx + (x)*Dx**2, x)
+    assert p*q == r
+
+
+def test_HolonomicFunction_power():
+    x = symbols('x')
+    R, Dx = DifferentialOperators(ZZ.old_poly_ring(x), 'Dx')
+    p = HolonomicFunction(Dx+x+x*Dx**2, x)
+    a = HolonomicFunction(Dx, x)
+    for n in range(10):
+        assert a == p**n
+        a *= p
+
+
+def test_addition_initial_condition():
+    x = symbols('x')
+    R, Dx = DifferentialOperators(QQ.old_poly_ring(x), 'Dx')
+    p = HolonomicFunction(Dx-1, x, 0, [3])
+    q = HolonomicFunction(Dx**2+1, x, 0, [1, 0])
+    r = HolonomicFunction(-1 + Dx - Dx**2 + Dx**3, x, 0, [4, 3, 2])
+    assert p + q == r
+    p = HolonomicFunction(Dx - x + Dx**2, x, 0, [1, 2])
+    q = HolonomicFunction(Dx**2 + x, x, 0, [1, 0])
+    r = HolonomicFunction((-x**4 - x**3/4 - x**2 + Rational(1, 4)) + (x**3 + x**2/4 + x*Rational(3, 4) + 1)*Dx + \
+        (x*Rational(-3, 2) + Rational(7, 4))*Dx**2 + (x**2 - x*Rational(7, 4) + Rational(1, 4))*Dx**3 + (x**2 + x/4 + S.Half)*Dx**4, x, 0, [2, 2, -2, 2])
+    assert p + q == r
+    p = HolonomicFunction(Dx**2 + 4*x*Dx + x**2, x, 0, [3, 4])
+    q = HolonomicFunction(Dx**2 + 1, x, 0, [1, 1])
+    r = HolonomicFunction((x**6 + 2*x**4 - 5*x**2 - 6) + (4*x**5 + 36*x**3 - 32*x)*Dx + \
+         (x**6 + 3*x**4 + 5*x**2 - 9)*Dx**2 + (4*x**5 + 36*x**3 - 32*x)*Dx**3 + (x**4 + \
+            10*x**2 - 3)*Dx**4, x, 0, [4, 5, -1, -17])
+    assert p + q == r
+    q = HolonomicFunction(Dx**3 + x, x, 2, [3, 0, 1])
+    p = HolonomicFunction(Dx - 1, x, 2, [1])
+    r = HolonomicFunction((-x**2 - x + 1) + (x**2 + x)*Dx + (-x - 2)*Dx**3 + \
+        (x + 1)*Dx**4, x, 2, [4, 1, 2, -5 ])
+    assert p + q == r
+    p = expr_to_holonomic(sin(x))
+    q = expr_to_holonomic(1/x, x0=1)
+    r = HolonomicFunction((x**2 + 6) + (x**3 + 2*x)*Dx + (x**2 + 6)*Dx**2 + (x**3 + 2*x)*Dx**3, \
+        x, 1, [sin(1) + 1, -1 + cos(1), -sin(1) + 2])
+    assert p + q == r
+    C_1 = symbols('C_1')
+    p = expr_to_holonomic(sqrt(x))
+    q = expr_to_holonomic(sqrt(x**2-x))
+    r = (p + q).to_expr().subs(C_1, -I/2).expand()
+    assert r == I*sqrt(x)*sqrt(-x + 1) + sqrt(x)
+
+
+def test_multiplication_initial_condition():
+    x = symbols('x')
+    R, Dx = DifferentialOperators(QQ.old_poly_ring(x), 'Dx')
+    p = HolonomicFunction(Dx**2 + x*Dx - 1, x, 0, [3, 1])
+    q = HolonomicFunction(Dx**2 + 1, x, 0, [1, 1])
+    r = HolonomicFunction((x**4 + 14*x**2 + 60) + 4*x*Dx + (x**4 + 9*x**2 + 20)*Dx**2 + \
+        (2*x**3 + 18*x)*Dx**3 + (x**2 + 10)*Dx**4, x, 0, [3, 4, 2, 3])
+    assert p * q == r
+    p = HolonomicFunction(Dx**2 + x, x, 0, [1, 0])
+    q = HolonomicFunction(Dx**3 - x**2, x, 0, [3, 3, 3])
+    r = HolonomicFunction((x**8 - 37*x**7/27 - 10*x**6/27 - 164*x**5/9 - 184*x**4/9 + \
+        160*x**3/27 + 404*x**2/9 + 8*x + Rational(40, 3)) + (6*x**7 - 128*x**6/9 - 98*x**5/9 - 28*x**4/9 + \
+        8*x**3/9 + 28*x**2 + x*Rational(40, 9) - 40)*Dx + (3*x**6 - 82*x**5/9 + 76*x**4/9 + 4*x**3/3 + \
+        220*x**2/9 - x*Rational(80, 3))*Dx**2 + (-2*x**6 + 128*x**5/27 - 2*x**4/3 -80*x**2/9 + Rational(200, 9))*Dx**3 + \
+        (3*x**5 - 64*x**4/9 - 28*x**3/9 + 6*x**2 - x*Rational(20, 9) - Rational(20, 3))*Dx**4 + (-4*x**3 + 64*x**2/9 + \
+            x*Rational(8, 3))*Dx**5 + (x**4 - 64*x**3/27 - 4*x**2/3 + Rational(20, 9))*Dx**6, x, 0, [3, 3, 3, -3, -12, -24])
+    assert p * q == r
+    p = HolonomicFunction(Dx - 1, x, 0, [2])
+    q = HolonomicFunction(Dx**2 + 1, x, 0, [0, 1])
+    r = HolonomicFunction(2 -2*Dx + Dx**2, x, 0, [0, 2])
+    assert p * q == r
+    q = HolonomicFunction(x*Dx**2 + 1 + 2*Dx, x, 0,[0, 1])
+    r = HolonomicFunction((x - 1) + (-2*x + 2)*Dx + x*Dx**2, x, 0, [0, 2])
+    assert p * q == r
+    p = HolonomicFunction(Dx**2 - 1, x, 0, [1, 3])
+    q = HolonomicFunction(Dx**3 + 1, x, 0, [1, 2, 1])
+    r = HolonomicFunction(6*Dx + 3*Dx**2 + 2*Dx**3 - 3*Dx**4 + Dx**6, x, 0, [1, 5, 14, 17, 17, 2])
+    assert p * q == r
+    p = expr_to_holonomic(sin(x))
+    q = expr_to_holonomic(1/x, x0=1)
+    r = HolonomicFunction(x + 2*Dx + x*Dx**2, x, 1, [sin(1), -sin(1) + cos(1)])
+    assert p * q == r
+    p = expr_to_holonomic(sqrt(x))
+    q = expr_to_holonomic(sqrt(x**2-x))
+    r = (p * q).to_expr()
+    assert r == I*x*sqrt(-x + 1)
+
+
+def test_HolonomicFunction_composition():
+    x = symbols('x')
+    R, Dx = DifferentialOperators(ZZ.old_poly_ring(x), 'Dx')
+    p = HolonomicFunction(Dx-1, x).composition(x**2+x)
+    r = HolonomicFunction((-2*x - 1) + Dx, x)
+    assert p == r
+    p = HolonomicFunction(Dx**2+1, x).composition(x**5+x**2+1)
+    r = HolonomicFunction((125*x**12 + 150*x**9 + 60*x**6 + 8*x**3) + (-20*x**3 - 2)*Dx + \
+        (5*x**4 + 2*x)*Dx**2, x)
+    assert p == r
+    p = HolonomicFunction(Dx**2*x+x, x).composition(2*x**3+x**2+1)
+    r = HolonomicFunction((216*x**9 + 324*x**8 + 180*x**7 + 152*x**6 + 112*x**5 + \
+        36*x**4 + 4*x**3) + (24*x**4 + 16*x**3 + 3*x**2 - 6*x - 1)*Dx + (6*x**5 + 5*x**4 + \
+        x**3 + 3*x**2 + x)*Dx**2, x)
+    assert p == r
+    p = HolonomicFunction(Dx**2+1, x).composition(1-x**2)
+    r = HolonomicFunction((4*x**3) - Dx + x*Dx**2, x)
+    assert p == r
+    p = HolonomicFunction(Dx**2+1, x).composition(x - 2/(x**2 + 1))
+    r = HolonomicFunction((x**12 + 6*x**10 + 12*x**9 + 15*x**8 + 48*x**7 + 68*x**6 + \
+        72*x**5 + 111*x**4 + 112*x**3 + 54*x**2 + 12*x + 1) + (12*x**8 + 32*x**6 + \
+        24*x**4 - 4)*Dx + (x**12 + 6*x**10 + 4*x**9 + 15*x**8 + 16*x**7 + 20*x**6 + 24*x**5+ \
+        15*x**4 + 16*x**3 + 6*x**2 + 4*x + 1)*Dx**2, x)
+    assert p == r
+
+
+def test_from_hyper():
+    x = symbols('x')
+    R, Dx = DifferentialOperators(QQ.old_poly_ring(x), 'Dx')
+    p = hyper([1, 1], [Rational(3, 2)], x**2/4)
+    q = HolonomicFunction((4*x) + (5*x**2 - 8)*Dx + (x**3 - 4*x)*Dx**2, x, 1, [2*sqrt(3)*pi/9, -4*sqrt(3)*pi/27 + Rational(4, 3)])
+    r = from_hyper(p)
+    assert r == q
+    p = from_hyper(hyper([1], [Rational(3, 2)], x**2/4))
+    q = HolonomicFunction(-x + (-x**2/2 + 2)*Dx + x*Dx**2, x)
+    # x0 = 1
+    y0 = '[sqrt(pi)*exp(1/4)*erf(1/2), -sqrt(pi)*exp(1/4)*erf(1/2)/2 + 1]'
+    assert sstr(p.y0) == y0
+    assert q.annihilator == p.annihilator
+
+
+def test_from_meijerg():
+    x = symbols('x')
+    R, Dx = DifferentialOperators(QQ.old_poly_ring(x), 'Dx')
+    p = from_meijerg(meijerg(([], [Rational(3, 2)]), ([S.Half], [S.Half, 1]), x))
+    q = HolonomicFunction(x/2 - Rational(1, 4) + (-x**2 + x/4)*Dx + x**2*Dx**2 + x**3*Dx**3, x, 1, \
+        [1/sqrt(pi), 1/(2*sqrt(pi)), -1/(4*sqrt(pi))])
+    assert p == q
+    p = from_meijerg(meijerg(([], []), ([0], []), x))
+    q = HolonomicFunction(1 + Dx, x, 0, [1])
+    assert p == q
+    p = from_meijerg(meijerg(([1], []), ([S.Half], [0]), x))
+    q = HolonomicFunction((x + S.Half)*Dx + x*Dx**2, x, 1, [sqrt(pi)*erf(1), exp(-1)])
+    assert p == q
+    p = from_meijerg(meijerg(([0], [1]), ([0], []), 2*x**2))
+    q = HolonomicFunction((3*x**2 - 1)*Dx + x**3*Dx**2, x, 1, [-exp(Rational(-1, 2)) + 1, -exp(Rational(-1, 2))])
+    assert p == q
+
+
+def test_to_Sequence():
+    x = symbols('x')
+    R, Dx = DifferentialOperators(ZZ.old_poly_ring(x), 'Dx')
+    n = symbols('n', integer=True)
+    _, Sn = RecurrenceOperators(ZZ.old_poly_ring(n), 'Sn')
+    p = HolonomicFunction(x**2*Dx**4 + x + Dx, x).to_sequence()
+    q = [(HolonomicSequence(1 + (n + 2)*Sn**2 + (n**4 + 6*n**3 + 11*n**2 + 6*n)*Sn**3), 0, 1)]
+    assert p == q
+    p = HolonomicFunction(x**2*Dx**4 + x**3 + Dx**2, x).to_sequence()
+    q = [(HolonomicSequence(1 + (n**4 + 14*n**3 + 72*n**2 + 163*n + 140)*Sn**5), 0, 0)]
+    assert p == q
+    p = HolonomicFunction(x**3*Dx**4 + 1 + Dx**2, x).to_sequence()
+    q = [(HolonomicSequence(1 + (n**4 - 2*n**3 - n**2 + 2*n)*Sn + (n**2 + 3*n + 2)*Sn**2), 0, 0)]
+    assert p == q
+    p = HolonomicFunction(3*x**3*Dx**4 + 2*x*Dx + x*Dx**3, x).to_sequence()
+    q = [(HolonomicSequence(2*n + (3*n**4 - 6*n**3 - 3*n**2 + 6*n)*Sn + (n**3 + 3*n**2 + 2*n)*Sn**2), 0, 1)]
+    assert p == q
+
+
+def test_to_Sequence_Initial_Coniditons():
+    x = symbols('x')
+    R, Dx = DifferentialOperators(QQ.old_poly_ring(x), 'Dx')
+    n = symbols('n', integer=True)
+    _, Sn = RecurrenceOperators(QQ.old_poly_ring(n), 'Sn')
+    p = HolonomicFunction(Dx - 1, x, 0, [1]).to_sequence()
+    q = [(HolonomicSequence(-1 + (n + 1)*Sn, 1), 0)]
+    assert p == q
+    p = HolonomicFunction(Dx**2 + 1, x, 0, [0, 1]).to_sequence()
+    q = [(HolonomicSequence(1 + (n**2 + 3*n + 2)*Sn**2, [0, 1]), 0)]
+    assert p == q
+    p = HolonomicFunction(Dx**2 + 1 + x**3*Dx, x, 0, [2, 3]).to_sequence()
+    q = [(HolonomicSequence(n + Sn**2 + (n**2 + 7*n + 12)*Sn**4, [2, 3, -1, Rational(-1, 2), Rational(1, 12)]), 1)]
+    assert p == q
+    p = HolonomicFunction(x**3*Dx**5 + 1 + Dx, x).to_sequence()
+    q = [(HolonomicSequence(1 + (n + 1)*Sn + (n**5 - 5*n**3 + 4*n)*Sn**2), 0, 3)]
+    assert p == q
+    C_0, C_1, C_2, C_3 = symbols('C_0, C_1, C_2, C_3')
+    p = expr_to_holonomic(log(1+x**2))
+    q = [(HolonomicSequence(n**2 + (n**2 + 2*n)*Sn**2, [0, 0, C_2]), 0, 1)]
+    assert p.to_sequence() == q
+    p = p.diff()
+    q = [(HolonomicSequence((n + 2) + (n + 2)*Sn**2, [C_0, 0]), 1, 0)]
+    assert p.to_sequence() == q
+    p = expr_to_holonomic(erf(x) + x).to_sequence()
+    q = [(HolonomicSequence((2*n**2 - 2*n) + (n**3 + 2*n**2 - n - 2)*Sn**2, [0, 1 + 2/sqrt(pi), 0, C_3]), 0, 2)]
+    assert p == q
+
+def test_series():
+    x = symbols('x')
+    R, Dx = DifferentialOperators(ZZ.old_poly_ring(x), 'Dx')
+    p = HolonomicFunction(Dx**2 + 2*x*Dx, x, 0, [0, 1]).series(n=10)
+    q = x - x**3/3 + x**5/10 - x**7/42 + x**9/216 + O(x**10)
+    assert p == q
+    p = HolonomicFunction(Dx - 1, x).composition(x**2, 0, [1])  # e^(x**2)
+    q = HolonomicFunction(Dx**2 + 1, x, 0, [1, 0])  # cos(x)
+    r = (p * q).series(n=10)  # expansion of cos(x) * exp(x**2)
+    s = 1 + x**2/2 + x**4/24 - 31*x**6/720 - 179*x**8/8064 + O(x**10)
+    assert r == s
+    t = HolonomicFunction((1 + x)*Dx**2 + Dx, x, 0, [0, 1])  # log(1 + x)
+    r = (p * t + q).series(n=10)
+    s = 1 + x - x**2 + 4*x**3/3 - 17*x**4/24 + 31*x**5/30 - 481*x**6/720 +\
+     71*x**7/105 - 20159*x**8/40320 + 379*x**9/840 + O(x**10)
+    assert r == s
+    p = HolonomicFunction((6+6*x-3*x**2) - (10*x-3*x**2-3*x**3)*Dx + \
+        (4-6*x**3+2*x**4)*Dx**2, x, 0, [0, 1]).series(n=7)
+    q = x + x**3/6 - 3*x**4/16 + x**5/20 - 23*x**6/960 + O(x**7)
+    assert p == q
+    p = HolonomicFunction((6+6*x-3*x**2) - (10*x-3*x**2-3*x**3)*Dx + \
+        (4-6*x**3+2*x**4)*Dx**2, x, 0, [1, 0]).series(n=7)
+    q = 1 - 3*x**2/4 - x**3/4 - 5*x**4/32 - 3*x**5/40 - 17*x**6/384 + O(x**7)
+    assert p == q
+    p = expr_to_holonomic(erf(x) + x).series(n=10)
+    C_3 = symbols('C_3')
+    q = (erf(x) + x).series(n=10)
+    assert p.subs(C_3, -2/(3*sqrt(pi))) == q
+    assert expr_to_holonomic(sqrt(x**3 + x)).series(n=10) == sqrt(x**3 + x).series(n=10)
+    assert expr_to_holonomic((2*x - 3*x**2)**Rational(1, 3)).series() == ((2*x - 3*x**2)**Rational(1, 3)).series()
+    assert  expr_to_holonomic(sqrt(x**2-x)).series() == (sqrt(x**2-x)).series()
+    assert expr_to_holonomic(cos(x)**2/x**2, y0={-2: [1, 0, -1]}).series(n=10) == (cos(x)**2/x**2).series(n=10)
+    assert expr_to_holonomic(cos(x)**2/x**2, x0=1).series(n=10).together() == (cos(x)**2/x**2).series(n=10, x0=1).together()
+    assert expr_to_holonomic(cos(x-1)**2/(x-1)**2, x0=1, y0={-2: [1, 0, -1]}).series(n=10) \
+        == (cos(x-1)**2/(x-1)**2).series(x0=1, n=10)
+
+def test_evalf_euler():
+    x = symbols('x')
+    R, Dx = DifferentialOperators(QQ.old_poly_ring(x), 'Dx')
+
+    # log(1+x)
+    p = HolonomicFunction((1 + x)*Dx**2 + Dx, x, 0, [0, 1])
+
+    # path taken is a straight line from 0 to 1, on the real axis
+    r = [0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1]
+    s = '0.699525841805253'  # approx. equal to log(2) i.e. 0.693147180559945
+    assert sstr(p.evalf(r, method='Euler')[-1]) == s
+
+    # path taken is a triangle 0-->1+i-->2
+    r = [0.1 + 0.1*I]
+    for i in range(9):
+        r.append(r[-1]+0.1+0.1*I)
+    for i in range(10):
+        r.append(r[-1]+0.1-0.1*I)
+
+    # close to the exact solution 1.09861228866811
+    # imaginary part also close to zero
+    s = '1.07530466271334 - 0.0251200594793912*I'
+    assert sstr(p.evalf(r, method='Euler')[-1]) == s
+
+    # sin(x)
+    p = HolonomicFunction(Dx**2 + 1, x, 0, [0, 1])
+    s = '0.905546532085401 - 6.93889390390723e-18*I'
+    assert sstr(p.evalf(r, method='Euler')[-1]) == s
+
+    # computing sin(pi/2) using this method
+    # using a linear path from 0 to pi/2
+    r = [0.1]
+    for i in range(14):
+        r.append(r[-1] + 0.1)
+    r.append(pi/2)
+    s = '1.08016557252834' # close to 1.0 (exact solution)
+    assert sstr(p.evalf(r, method='Euler')[-1]) == s
+
+    # trying different path, a rectangle (0-->i-->pi/2 + i-->pi/2)
+    # computing the same value sin(pi/2) using different path
+    r = [0.1*I]
+    for i in range(9):
+        r.append(r[-1]+0.1*I)
+    for i in range(15):
+        r.append(r[-1]+0.1)
+    r.append(pi/2+I)
+    for i in range(10):
+        r.append(r[-1]-0.1*I)
+
+    # close to 1.0
+    s = '0.976882381836257 - 1.65557671738537e-16*I'
+    assert sstr(p.evalf(r, method='Euler')[-1]) == s
+
+    # cos(x)
+    p = HolonomicFunction(Dx**2 + 1, x, 0, [1, 0])
+    # compute cos(pi) along 0-->pi
+    r = [0.05]
+    for i in range(61):
+        r.append(r[-1]+0.05)
+    r.append(pi)
+    # close to -1 (exact answer)
+    s = '-1.08140824719196'
+    assert sstr(p.evalf(r, method='Euler')[-1]) == s
+
+    # a rectangular path (0 -> i -> 2+i -> 2)
+    r = [0.1*I]
+    for i in range(9):
+        r.append(r[-1]+0.1*I)
+    for i in range(20):
+        r.append(r[-1]+0.1)
+    for i in range(10):
+        r.append(r[-1]-0.1*I)
+
+    p = HolonomicFunction(Dx**2 + 1, x, 0, [1,1]).evalf(r, method='Euler')
+    s = '0.501421652861245 - 3.88578058618805e-16*I'
+    assert sstr(p[-1]) == s
+
+def test_evalf_rk4():
+    x = symbols('x')
+    R, Dx = DifferentialOperators(QQ.old_poly_ring(x), 'Dx')
+
+    # log(1+x)
+    p = HolonomicFunction((1 + x)*Dx**2 + Dx, x, 0, [0, 1])
+
+    # path taken is a straight line from 0 to 1, on the real axis
+    r = [0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1]
+    s = '0.693146363174626'  # approx. equal to log(2) i.e. 0.693147180559945
+    assert sstr(p.evalf(r)[-1]) == s
+
+    # path taken is a triangle 0-->1+i-->2
+    r = [0.1 + 0.1*I]
+    for i in range(9):
+        r.append(r[-1]+0.1+0.1*I)
+    for i in range(10):
+        r.append(r[-1]+0.1-0.1*I)
+
+    # close to the exact solution 1.09861228866811
+    # imaginary part also close to zero
+    s = '1.098616 + 1.36083e-7*I'
+    assert sstr(p.evalf(r)[-1].n(7)) == s
+
+    # sin(x)
+    p = HolonomicFunction(Dx**2 + 1, x, 0, [0, 1])
+    s = '0.90929463522785 + 1.52655665885959e-16*I'
+    assert sstr(p.evalf(r)[-1]) == s
+
+    # computing sin(pi/2) using this method
+    # using a linear path from 0 to pi/2
+    r = [0.1]
+    for i in range(14):
+        r.append(r[-1] + 0.1)
+    r.append(pi/2)
+    s = '0.999999895088917' # close to 1.0 (exact solution)
+    assert sstr(p.evalf(r)[-1]) == s
+
+    # trying different path, a rectangle (0-->i-->pi/2 + i-->pi/2)
+    # computing the same value sin(pi/2) using different path
+    r = [0.1*I]
+    for i in range(9):
+        r.append(r[-1]+0.1*I)
+    for i in range(15):
+        r.append(r[-1]+0.1)
+    r.append(pi/2+I)
+    for i in range(10):
+        r.append(r[-1]-0.1*I)
+
+    # close to 1.0
+    s = '1.00000003415141 + 6.11940487991086e-16*I'
+    assert sstr(p.evalf(r)[-1]) == s
+
+    # cos(x)
+    p = HolonomicFunction(Dx**2 + 1, x, 0, [1, 0])
+    # compute cos(pi) along 0-->pi
+    r = [0.05]
+    for i in range(61):
+        r.append(r[-1]+0.05)
+    r.append(pi)
+    # close to -1 (exact answer)
+    s = '-0.999999993238714'
+    assert sstr(p.evalf(r)[-1]) == s
+
+    # a rectangular path (0 -> i -> 2+i -> 2)
+    r = [0.1*I]
+    for i in range(9):
+        r.append(r[-1]+0.1*I)
+    for i in range(20):
+        r.append(r[-1]+0.1)
+    for i in range(10):
+        r.append(r[-1]-0.1*I)
+
+    p = HolonomicFunction(Dx**2 + 1, x, 0, [1,1]).evalf(r)
+    s = '0.493152791638442 - 1.41553435639707e-15*I'
+    assert sstr(p[-1]) == s
+
+
+def test_expr_to_holonomic():
+    x = symbols('x')
+    R, Dx = DifferentialOperators(QQ.old_poly_ring(x), 'Dx')
+    p = expr_to_holonomic((sin(x)/x)**2)
+    q = HolonomicFunction(8*x + (4*x**2 + 6)*Dx + 6*x*Dx**2 + x**2*Dx**3, x, 0, \
+        [1, 0, Rational(-2, 3)])
+    assert p == q
+    p = expr_to_holonomic(1/(1+x**2)**2)
+    q = HolonomicFunction(4*x + (x**2 + 1)*Dx, x, 0, [1])
+    assert p == q
+    p = expr_to_holonomic(exp(x)*sin(x)+x*log(1+x))
+    q = HolonomicFunction((2*x**3 + 10*x**2 + 20*x + 18) + (-2*x**4 - 10*x**3 - 20*x**2 \
+        - 18*x)*Dx + (2*x**5 + 6*x**4 + 7*x**3 + 8*x**2 + 10*x - 4)*Dx**2 + \
+        (-2*x**5 - 5*x**4 - 2*x**3 + 2*x**2 - x + 4)*Dx**3 + (x**5 + 2*x**4 - x**3 - \
+        7*x**2/2 + x + Rational(5, 2))*Dx**4, x, 0, [0, 1, 4, -1])
+    assert p == q
+    p = expr_to_holonomic(x*exp(x)+cos(x)+1)
+    q = HolonomicFunction((-x - 3)*Dx + (x + 2)*Dx**2 + (-x - 3)*Dx**3 + (x + 2)*Dx**4, x, \
+        0, [2, 1, 1, 3])
+    assert p == q
+    assert (x*exp(x)+cos(x)+1).series(n=10) == p.series(n=10)
+    p = expr_to_holonomic(log(1 + x)**2 + 1)
+    q = HolonomicFunction(Dx + (3*x + 3)*Dx**2 + (x**2 + 2*x + 1)*Dx**3, x, 0, [1, 0, 2])
+    assert p == q
+    p = expr_to_holonomic(erf(x)**2 + x)
+    q = HolonomicFunction((8*x**4 - 2*x**2 + 2)*Dx**2 + (6*x**3 - x/2)*Dx**3 + \
+        (x**2+ Rational(1, 4))*Dx**4, x, 0, [0, 1, 8/pi, 0])
+    assert p == q
+    p = expr_to_holonomic(cosh(x)*x)
+    q = HolonomicFunction((-x**2 + 2) -2*x*Dx + x**2*Dx**2, x, 0, [0, 1])
+    assert p == q
+    p = expr_to_holonomic(besselj(2, x))
+    q = HolonomicFunction((x**2 - 4) + x*Dx + x**2*Dx**2, x, 0, [0, 0])
+    assert p == q
+    p = expr_to_holonomic(besselj(0, x) + exp(x))
+    q = HolonomicFunction((-x**2 - x/2 + S.Half) + (x**2 - x/2 - Rational(3, 2))*Dx + (-x**2 + x/2 + 1)*Dx**2 +\
+        (x**2 + x/2)*Dx**3, x, 0, [2, 1, S.Half])
+    assert p == q
+    p = expr_to_holonomic(sin(x)**2/x)
+    q = HolonomicFunction(4 + 4*x*Dx + 3*Dx**2 + x*Dx**3, x, 0, [0, 1, 0])
+    assert p == q
+    p = expr_to_holonomic(sin(x)**2/x, x0=2)
+    q = HolonomicFunction((4) + (4*x)*Dx + (3)*Dx**2 + (x)*Dx**3, x, 2, [sin(2)**2/2,
+        sin(2)*cos(2) - sin(2)**2/4, -3*sin(2)**2/4 + cos(2)**2 - sin(2)*cos(2)])
+    assert p == q
+    p = expr_to_holonomic(log(x)/2 - Ci(2*x)/2 + Ci(2)/2)
+    q = HolonomicFunction(4*Dx + 4*x*Dx**2 + 3*Dx**3 + x*Dx**4, x, 0, \
+        [-log(2)/2 - EulerGamma/2 + Ci(2)/2, 0, 1, 0])
+    assert p == q
+    p = p.to_expr()
+    q = log(x)/2 - Ci(2*x)/2 + Ci(2)/2
+    assert p == q
+    p = expr_to_holonomic(x**S.Half, x0=1)
+    q = HolonomicFunction(x*Dx - S.Half, x, 1, [1])
+    assert p == q
+    p = expr_to_holonomic(sqrt(1 + x**2))
+    q = HolonomicFunction((-x) + (x**2 + 1)*Dx, x, 0, [1])
+    assert p == q
+    assert (expr_to_holonomic(sqrt(x) + sqrt(2*x)).to_expr()-\
+        (sqrt(x) + sqrt(2*x))).simplify() == 0
+    assert expr_to_holonomic(3*x+2*sqrt(x)).to_expr() == 3*x+2*sqrt(x)
+    p = expr_to_holonomic((x**4+x**3+5*x**2+3*x+2)/x**2, lenics=3)
+    q = HolonomicFunction((-2*x**4 - x**3 + 3*x + 4) + (x**5 + x**4 + 5*x**3 + 3*x**2 + \
+        2*x)*Dx, x, 0, {-2: [2, 3, 5]})
+    assert p == q
+    p = expr_to_holonomic(1/(x-1)**2, lenics=3, x0=1)
+    q = HolonomicFunction((2) + (x - 1)*Dx, x, 1, {-2: [1, 0, 0]})
+    assert p == q
+    a = symbols("a")
+    p = expr_to_holonomic(sqrt(a*x), x=x)
+    assert p.to_expr() == sqrt(a)*sqrt(x)
+
+def test_to_hyper():
+    x = symbols('x')
+    R, Dx = DifferentialOperators(QQ.old_poly_ring(x), 'Dx')
+    p = HolonomicFunction(Dx - 2, x, 0, [3]).to_hyper()
+    q = 3 * hyper([], [], 2*x)
+    assert p == q
+    p = hyperexpand(HolonomicFunction((1 + x) * Dx - 3, x, 0, [2]).to_hyper()).expand()
+    q = 2*x**3 + 6*x**2 + 6*x + 2
+    assert p == q
+    p = HolonomicFunction((1 + x)*Dx**2 + Dx, x, 0, [0, 1]).to_hyper()
+    q = -x**2*hyper((2, 2, 1), (3, 2), -x)/2 + x
+    assert p == q
+    p = HolonomicFunction(2*x*Dx + Dx**2, x, 0, [0, 2/sqrt(pi)]).to_hyper()
+    q = 2*x*hyper((S.Half,), (Rational(3, 2),), -x**2)/sqrt(pi)
+    assert p == q
+    p = hyperexpand(HolonomicFunction(2*x*Dx + Dx**2, x, 0, [1, -2/sqrt(pi)]).to_hyper())
+    q = erfc(x)
+    assert p.rewrite(erfc) == q
+    p =  hyperexpand(HolonomicFunction((x**2 - 1) + x*Dx + x**2*Dx**2,
+        x, 0, [0, S.Half]).to_hyper())
+    q = besselj(1, x)
+    assert p == q
+    p = hyperexpand(HolonomicFunction(x*Dx**2 + Dx + x, x, 0, [1, 0]).to_hyper())
+    q = besselj(0, x)
+    assert p == q
+
+def test_to_expr():
+    x = symbols('x')
+    R, Dx = DifferentialOperators(ZZ.old_poly_ring(x), 'Dx')
+    p = HolonomicFunction(Dx - 1, x, 0, [1]).to_expr()
+    q = exp(x)
+    assert p == q
+    p = HolonomicFunction(Dx**2 + 1, x, 0, [1, 0]).to_expr()
+    q = cos(x)
+    assert p == q
+    p = HolonomicFunction(Dx**2 - 1, x, 0, [1, 0]).to_expr()
+    q = cosh(x)
+    assert p == q
+    p = HolonomicFunction(2 + (4*x - 1)*Dx + \
+        (x**2 - x)*Dx**2, x, 0, [1, 2]).to_expr().expand()
+    q = 1/(x**2 - 2*x + 1)
+    assert p == q
+    p = expr_to_holonomic(sin(x)**2/x).integrate((x, 0, x)).to_expr()
+    q = (sin(x)**2/x).integrate((x, 0, x))
+    assert p == q
+    C_0, C_1, C_2, C_3 = symbols('C_0, C_1, C_2, C_3')
+    p = expr_to_holonomic(log(1+x**2)).to_expr()
+    q = C_2*log(x**2 + 1)
+    assert p == q
+    p = expr_to_holonomic(log(1+x**2)).diff().to_expr()
+    q = C_0*x/(x**2 + 1)
+    assert p == q
+    p = expr_to_holonomic(erf(x) + x).to_expr()
+    q = 3*C_3*x - 3*sqrt(pi)*C_3*erf(x)/2 + x + 2*x/sqrt(pi)
+    assert p == q
+    p = expr_to_holonomic(sqrt(x), x0=1).to_expr()
+    assert p == sqrt(x)
+    assert expr_to_holonomic(sqrt(x)).to_expr() == sqrt(x)
+    p = expr_to_holonomic(sqrt(1 + x**2)).to_expr()
+    assert p == sqrt(1+x**2)
+    p = expr_to_holonomic((2*x**2 + 1)**Rational(2, 3)).to_expr()
+    assert p == (2*x**2 + 1)**Rational(2, 3)
+    p = expr_to_holonomic(sqrt(-x**2+2*x)).to_expr()
+    assert p == sqrt(x)*sqrt(-x + 2)
+    p = expr_to_holonomic((-2*x**3+7*x)**Rational(2, 3)).to_expr()
+    q = x**Rational(2, 3)*(-2*x**2 + 7)**Rational(2, 3)
+    assert p == q
+    p = from_hyper(hyper((-2, -3), (S.Half, ), x))
+    s = hyperexpand(hyper((-2, -3), (S.Half, ), x))
+    D_0 = Symbol('D_0')
+    C_0 = Symbol('C_0')
+    assert (p.to_expr().subs({C_0:1, D_0:0}) - s).simplify() == 0
+    p.y0 = {0: [1], S.Half: [0]}
+    assert p.to_expr() == s
+    assert expr_to_holonomic(x**5).to_expr() == x**5
+    assert expr_to_holonomic(2*x**3-3*x**2).to_expr().expand() == \
+        2*x**3-3*x**2
+    a = symbols("a")
+    p = (expr_to_holonomic(1.4*x)*expr_to_holonomic(a*x, x)).to_expr()
+    q = 1.4*a*x**2
+    assert p == q
+    p = (expr_to_holonomic(1.4*x)+expr_to_holonomic(a*x, x)).to_expr()
+    q = x*(a + 1.4)
+    assert p == q
+    p = (expr_to_holonomic(1.4*x)+expr_to_holonomic(x)).to_expr()
+    assert p == 2.4*x
+
+
+def test_integrate():
+    x = symbols('x')
+    R, Dx = DifferentialOperators(ZZ.old_poly_ring(x), 'Dx')
+    p = expr_to_holonomic(sin(x)**2/x, x0=1).integrate((x, 2, 3))
+    q = '0.166270406994788'
+    assert sstr(p) == q
+    p = expr_to_holonomic(sin(x)).integrate((x, 0, x)).to_expr()
+    q = 1 - cos(x)
+    assert p == q
+    p = expr_to_holonomic(sin(x)).integrate((x, 0, 3))
+    q = 1 - cos(3)
+    assert p == q
+    p = expr_to_holonomic(sin(x)/x, x0=1).integrate((x, 1, 2))
+    q = '0.659329913368450'
+    assert sstr(p) == q
+    p = expr_to_holonomic(sin(x)**2/x, x0=1).integrate((x, 1, 0))
+    q = '-0.423690480850035'
+    assert sstr(p) == q
+    p = expr_to_holonomic(sin(x)/x)
+    assert p.integrate(x).to_expr() == Si(x)
+    assert p.integrate((x, 0, 2)) == Si(2)
+    p = expr_to_holonomic(sin(x)**2/x)
+    q = p.to_expr()
+    assert p.integrate(x).to_expr() == q.integrate((x, 0, x))
+    assert p.integrate((x, 0, 1)) == q.integrate((x, 0, 1))
+    assert expr_to_holonomic(1/x, x0=1).integrate(x).to_expr() == log(x)
+    p = expr_to_holonomic((x + 1)**3*exp(-x), x0=-1).integrate(x).to_expr()
+    q = (-x**3 - 6*x**2 - 15*x + 6*exp(x + 1) - 16)*exp(-x)
+    assert p == q
+    p = expr_to_holonomic(cos(x)**2/x**2, y0={-2: [1, 0, -1]}).integrate(x).to_expr()
+    q = -Si(2*x) - cos(x)**2/x
+    assert p == q
+    p = expr_to_holonomic(sqrt(x**2+x)).integrate(x).to_expr()
+    q = (x**Rational(3, 2)*(2*x**2 + 3*x + 1) - x*sqrt(x + 1)*asinh(sqrt(x)))/(4*x*sqrt(x + 1))
+    assert p == q
+    p = expr_to_holonomic(sqrt(x**2+1)).integrate(x).to_expr()
+    q = (sqrt(x**2+1)).integrate(x)
+    assert (p-q).simplify() == 0
+    p = expr_to_holonomic(1/x**2, y0={-2:[1, 0, 0]})
+    r = expr_to_holonomic(1/x**2, lenics=3)
+    assert p == r
+    q = expr_to_holonomic(cos(x)**2)
+    assert (r*q).integrate(x).to_expr() == -Si(2*x) - cos(x)**2/x
+
+
+def test_diff():
+    x, y = symbols('x, y')
+    R, Dx = DifferentialOperators(ZZ.old_poly_ring(x), 'Dx')
+    p = HolonomicFunction(x*Dx**2 + 1, x, 0, [0, 1])
+    assert p.diff().to_expr() == p.to_expr().diff().simplify()
+    p = HolonomicFunction(Dx**2 - 1, x, 0, [1, 0])
+    assert p.diff(x, 2).to_expr() == p.to_expr()
+    p = expr_to_holonomic(Si(x))
+    assert p.diff().to_expr() == sin(x)/x
+    assert p.diff(y) == 0
+    C_0, C_1, C_2, C_3 = symbols('C_0, C_1, C_2, C_3')
+    q = Si(x)
+    assert p.diff(x).to_expr() == q.diff()
+    assert p.diff(x, 2).to_expr().subs(C_0, Rational(-1, 3)).cancel() == q.diff(x, 2).cancel()
+    assert p.diff(x, 3).series().subs({C_3: Rational(-1, 3), C_0: 0}) == q.diff(x, 3).series()
+
+
+def test_extended_domain_in_expr_to_holonomic():
+    x = symbols('x')
+    p = expr_to_holonomic(1.2*cos(3.1*x))
+    assert p.to_expr() == 1.2*cos(3.1*x)
+    assert sstr(p.integrate(x).to_expr()) == '0.387096774193548*sin(3.1*x)'
+    _, Dx = DifferentialOperators(RR.old_poly_ring(x), 'Dx')
+    p = expr_to_holonomic(1.1329138213*x)
+    q = HolonomicFunction((-1.1329138213) + (1.1329138213*x)*Dx, x, 0, {1: [1.1329138213]})
+    assert p == q
+    assert p.to_expr() == 1.1329138213*x
+    assert sstr(p.integrate((x, 1, 2))) == sstr((1.1329138213*x).integrate((x, 1, 2)))
+    y, z = symbols('y, z')
+    p = expr_to_holonomic(sin(x*y*z), x=x)
+    assert p.to_expr() == sin(x*y*z)
+    assert p.integrate(x).to_expr() == (-cos(x*y*z) + 1)/(y*z)
+    p = expr_to_holonomic(sin(x*y + z), x=x).integrate(x).to_expr()
+    q = (cos(z) - cos(x*y + z))/y
+    assert p == q
+    a = symbols('a')
+    p = expr_to_holonomic(a*x, x)
+    assert p.to_expr() == a*x
+    assert p.integrate(x).to_expr() == a*x**2/2
+    D_2, C_1 = symbols("D_2, C_1")
+    p = expr_to_holonomic(x) + expr_to_holonomic(1.2*cos(x))
+    p = p.to_expr().subs(D_2, 0)
+    assert p - x - 1.2*cos(1.0*x) == 0
+    p = expr_to_holonomic(x) * expr_to_holonomic(1.2*cos(x))
+    p = p.to_expr().subs(C_1, 0)
+    assert p - 1.2*x*cos(1.0*x) == 0
+
+
+def test_to_meijerg():
+    x = symbols('x')
+    assert hyperexpand(expr_to_holonomic(sin(x)).to_meijerg()) == sin(x)
+    assert hyperexpand(expr_to_holonomic(cos(x)).to_meijerg()) == cos(x)
+    assert hyperexpand(expr_to_holonomic(exp(x)).to_meijerg()) == exp(x)
+    assert hyperexpand(expr_to_holonomic(log(x)).to_meijerg()).simplify() == log(x)
+    assert expr_to_holonomic(4*x**2/3 + 7).to_meijerg() == 4*x**2/3 + 7
+    assert hyperexpand(expr_to_holonomic(besselj(2, x), lenics=3).to_meijerg()) == besselj(2, x)
+    p = hyper((Rational(-1, 2), -3), (), x)
+    assert from_hyper(p).to_meijerg() == hyperexpand(p)
+    p = hyper((S.One, S(3)), (S(2), ), x)
+    assert (hyperexpand(from_hyper(p).to_meijerg()) - hyperexpand(p)).expand() == 0
+    p = from_hyper(hyper((-2, -3), (S.Half, ), x))
+    s = hyperexpand(hyper((-2, -3), (S.Half, ), x))
+    C_0 = Symbol('C_0')
+    C_1 = Symbol('C_1')
+    D_0 = Symbol('D_0')
+    assert (hyperexpand(p.to_meijerg()).subs({C_0:1, D_0:0}) - s).simplify() == 0
+    p.y0 = {0: [1], S.Half: [0]}
+    assert (hyperexpand(p.to_meijerg()) - s).simplify() == 0
+    p = expr_to_holonomic(besselj(S.Half, x), initcond=False)
+    assert (p.to_expr() - (D_0*sin(x) + C_0*cos(x) + C_1*sin(x))/sqrt(x)).simplify() == 0
+    p = expr_to_holonomic(besselj(S.Half, x), y0={Rational(-1, 2): [sqrt(2)/sqrt(pi), sqrt(2)/sqrt(pi)]})
+    assert (p.to_expr() - besselj(S.Half, x) - besselj(Rational(-1, 2), x)).simplify() == 0
+
+
+def test_gaussian():
+    mu, x = symbols("mu x")
+    sd = symbols("sd", positive=True)
+    Q = QQ[mu, sd].get_field()
+    e = sqrt(2)*exp(-(-mu + x)**2/(2*sd**2))/(2*sqrt(pi)*sd)
+    h1 = expr_to_holonomic(e, x, domain=Q)
+
+    _, Dx = DifferentialOperators(Q.old_poly_ring(x), 'Dx')
+    h2 = HolonomicFunction((-mu/sd**2 + x/sd**2) + (1)*Dx, x)
+
+    assert h1 == h2
+
+
+def test_beta():
+    a, b, x = symbols("a b x", positive=True)
+    e = x**(a - 1)*(-x + 1)**(b - 1)/beta(a, b)
+    Q = QQ[a, b].get_field()
+    h1 = expr_to_holonomic(e, x, domain=Q)
+
+    _, Dx = DifferentialOperators(Q.old_poly_ring(x), 'Dx')
+    h2 = HolonomicFunction((a + x*(-a - b + 2) - 1) + (x**2 - x)*Dx, x)
+
+    assert h1 == h2
+
+
+def test_gamma():
+    a, b, x = symbols("a b x", positive=True)
+    e = b**(-a)*x**(a - 1)*exp(-x/b)/gamma(a)
+    Q = QQ[a, b].get_field()
+    h1 = expr_to_holonomic(e, x, domain=Q)
+
+    _, Dx = DifferentialOperators(Q.old_poly_ring(x), 'Dx')
+    h2 = HolonomicFunction((-a + 1 + x/b) + (x)*Dx, x)
+
+    assert h1 == h2
+
+
+def test_symbolic_power():
+    x, n = symbols("x n")
+    Q = QQ[n].get_field()
+    _, Dx = DifferentialOperators(Q.old_poly_ring(x), 'Dx')
+    h1 = HolonomicFunction((-1) + (x)*Dx, x) ** -n
+    h2 = HolonomicFunction((n) + (x)*Dx, x)
+
+    assert h1 == h2
+
+
+def test_negative_power():
+    x = symbols("x")
+    _, Dx = DifferentialOperators(QQ.old_poly_ring(x), 'Dx')
+    h1 = HolonomicFunction((-1) + (x)*Dx, x) ** -2
+    h2 = HolonomicFunction((2) + (x)*Dx, x)
+
+    assert h1 == h2
+
+
+def test_expr_in_power():
+    x, n = symbols("x n")
+    Q = QQ[n].get_field()
+    _, Dx = DifferentialOperators(Q.old_poly_ring(x), 'Dx')
+    h1 = HolonomicFunction((-1) + (x)*Dx, x) ** (n - 3)
+    h2 = HolonomicFunction((-n + 3) + (x)*Dx, x)
+
+    assert h1 == h2
+
+
+def test_DifferentialOperatorEqPoly():
+    x = symbols('x', integer=True)
+    R, Dx = DifferentialOperators(QQ.old_poly_ring(x), 'Dx')
+    do = DifferentialOperator([x**2, R.base.zero, R.base.zero], R)
+    do2 = DifferentialOperator([x**2, 1, x], R)
+    assert not do == do2
+
+    # polynomial comparison issue, see https://github.com/sympy/sympy/pull/15799
+    # should work once that is solved
+    # p = do.listofpoly[0]
+    # assert do == p
+
+    p2 = do2.listofpoly[0]
+    assert not do2 == p2
+
+
+def test_DifferentialOperatorPow():
+    x = symbols('x', integer=True)
+    R, _ = DifferentialOperators(QQ.old_poly_ring(x), 'Dx')
+    do = DifferentialOperator([x**2, R.base.zero, R.base.zero], R)
+    a = DifferentialOperator([R.base.one], R)
+    for n in range(10):
+        assert a == do**n
+        a *= do

.venv/lib/python3.13/site-packages/sympy/holonomic/tests/test_recurrence.py

@@ -0,0 +1,41 @@
+from sympy.holonomic.recurrence import RecurrenceOperators, RecurrenceOperator
+from sympy.core.symbol import symbols
+from sympy.polys.domains.rationalfield import QQ
+
+
+def test_RecurrenceOperator():
+    n = symbols('n', integer=True)
+    R, Sn = RecurrenceOperators(QQ.old_poly_ring(n), 'Sn')
+    assert Sn*n == (n + 1)*Sn
+    assert Sn*n**2 == (n**2+1+2*n)*Sn
+    assert Sn**2*n**2 == (n**2 + 4*n + 4)*Sn**2
+    p = (Sn**3*n**2 + Sn*n)**2
+    q = (n**2 + 3*n + 2)*Sn**2 + (2*n**3 + 19*n**2 + 57*n + 52)*Sn**4 + (n**4 + 18*n**3 + \
+        117*n**2 + 324*n + 324)*Sn**6
+    assert p == q
+
+
+def test_RecurrenceOperatorEqPoly():
+    n = symbols('n', integer=True)
+    R, Sn = RecurrenceOperators(QQ.old_poly_ring(n), 'Sn')
+    rr = RecurrenceOperator([n**2, 0, 0], R)
+    rr2 = RecurrenceOperator([n**2, 1, n], R)
+    assert not rr == rr2
+
+    # polynomial comparison issue, see https://github.com/sympy/sympy/pull/15799
+    # should work once that is solved
+    # d = rr.listofpoly[0]
+    # assert rr == d
+
+    d2 = rr2.listofpoly[0]
+    assert not rr2 == d2
+
+
+def test_RecurrenceOperatorPow():
+    n = symbols('n', integer=True)
+    R, _ = RecurrenceOperators(QQ.old_poly_ring(n), 'Sn')
+    rr = RecurrenceOperator([n**2, 0, 0], R)
+    a = RecurrenceOperator([R.base.one], R)
+    for m in range(10):
+        assert a == rr**m
+        a *= rr

.venv/lib/python3.13/site-packages/sympy/polys/domains/__init__.py

@@ -0,0 +1,57 @@
+"""Implementation of mathematical domains. """
+
+__all__ = [
+    'Domain', 'FiniteField', 'IntegerRing', 'RationalField', 'RealField',
+    'ComplexField', 'AlgebraicField', 'PolynomialRing', 'FractionField',
+    'ExpressionDomain', 'PythonRational',
+
+    'GF', 'FF', 'ZZ', 'QQ', 'ZZ_I', 'QQ_I', 'RR', 'CC', 'EX', 'EXRAW',
+]
+
+from .domain import Domain
+from .finitefield import FiniteField, FF, GF
+from .integerring import IntegerRing, ZZ
+from .rationalfield import RationalField, QQ
+from .algebraicfield import AlgebraicField
+from .gaussiandomains import ZZ_I, QQ_I
+from .realfield import RealField, RR
+from .complexfield import ComplexField, CC
+from .polynomialring import PolynomialRing
+from .fractionfield import FractionField
+from .expressiondomain import ExpressionDomain, EX
+from .expressionrawdomain import EXRAW
+from .pythonrational import PythonRational
+
+
+# This is imported purely for backwards compatibility because some parts of
+# the codebase used to import this from here and it's possible that downstream
+# does as well:
+from sympy.external.gmpy import GROUND_TYPES  # noqa: F401
+
+#
+# The rest of these are obsolete and provided only for backwards
+# compatibility:
+#
+
+from .pythonfinitefield import PythonFiniteField
+from .gmpyfinitefield import GMPYFiniteField
+from .pythonintegerring import PythonIntegerRing
+from .gmpyintegerring import GMPYIntegerRing
+from .pythonrationalfield import PythonRationalField
+from .gmpyrationalfield import GMPYRationalField
+
+FF_python = PythonFiniteField
+FF_gmpy = GMPYFiniteField
+
+ZZ_python = PythonIntegerRing
+ZZ_gmpy = GMPYIntegerRing
+
+QQ_python = PythonRationalField
+QQ_gmpy = GMPYRationalField
+
+__all__.extend((
+    'PythonFiniteField', 'GMPYFiniteField', 'PythonIntegerRing',
+    'GMPYIntegerRing', 'PythonRational', 'GMPYRationalField',
+
+    'FF_python', 'FF_gmpy', 'ZZ_python', 'ZZ_gmpy', 'QQ_python', 'QQ_gmpy',
+))

.venv/lib/python3.13/site-packages/sympy/polys/domains/algebraicfield.py

@@ -0,0 +1,638 @@
+"""Implementation of :class:`AlgebraicField` class. """
+
+
+from sympy.core.add import Add
+from sympy.core.mul import Mul
+from sympy.core.singleton import S
+from sympy.core.symbol import Dummy, symbols
+from sympy.polys.domains.characteristiczero import CharacteristicZero
+from sympy.polys.domains.field import Field
+from sympy.polys.domains.simpledomain import SimpleDomain
+from sympy.polys.polyclasses import ANP
+from sympy.polys.polyerrors import CoercionFailed, DomainError, NotAlgebraic, IsomorphismFailed
+from sympy.utilities import public
+
+@public
+class AlgebraicField(Field, CharacteristicZero, SimpleDomain):
+    r"""Algebraic number field :ref:`QQ(a)`
+
+    A :ref:`QQ(a)` domain represents an `algebraic number field`_
+    `\mathbb{Q}(a)` as a :py:class:`~.Domain` in the domain system (see
+    :ref:`polys-domainsintro`).
+
+    A :py:class:`~.Poly` created from an expression involving `algebraic
+    numbers`_ will treat the algebraic numbers as generators if the generators
+    argument is not specified.
+
+    >>> from sympy import Poly, Symbol, sqrt
+    >>> x = Symbol('x')
+    >>> Poly(x**2 + sqrt(2))
+    Poly(x**2 + (sqrt(2)), x, sqrt(2), domain='ZZ')
+
+    That is a multivariate polynomial with ``sqrt(2)`` treated as one of the
+    generators (variables). If the generators are explicitly specified then
+    ``sqrt(2)`` will be considered to be a coefficient but by default the
+    :ref:`EX` domain is used. To make a :py:class:`~.Poly` with a :ref:`QQ(a)`
+    domain the argument ``extension=True`` can be given.
+
+    >>> Poly(x**2 + sqrt(2), x)
+    Poly(x**2 + sqrt(2), x, domain='EX')
+    >>> Poly(x**2 + sqrt(2), x, extension=True)
+    Poly(x**2 + sqrt(2), x, domain='QQ<sqrt(2)>')
+
+    A generator of the algebraic field extension can also be specified
+    explicitly which is particularly useful if the coefficients are all
+    rational but an extension field is needed (e.g. to factor the
+    polynomial).
+
+    >>> Poly(x**2 + 1)
+    Poly(x**2 + 1, x, domain='ZZ')
+    >>> Poly(x**2 + 1, extension=sqrt(2))
+    Poly(x**2 + 1, x, domain='QQ<sqrt(2)>')
+
+    It is possible to factorise a polynomial over a :ref:`QQ(a)` domain using
+    the ``extension`` argument to :py:func:`~.factor` or by specifying the domain
+    explicitly.
+
+    >>> from sympy import factor, QQ
+    >>> factor(x**2 - 2)
+    x**2 - 2
+    >>> factor(x**2 - 2, extension=sqrt(2))
+    (x - sqrt(2))*(x + sqrt(2))
+    >>> factor(x**2 - 2, domain='QQ<sqrt(2)>')
+    (x - sqrt(2))*(x + sqrt(2))
+    >>> factor(x**2 - 2, domain=QQ.algebraic_field(sqrt(2)))
+    (x - sqrt(2))*(x + sqrt(2))
+
+    The ``extension=True`` argument can be used but will only create an
+    extension that contains the coefficients which is usually not enough to
+    factorise the polynomial.
+
+    >>> p = x**3 + sqrt(2)*x**2 - 2*x - 2*sqrt(2)
+    >>> factor(p)                         # treats sqrt(2) as a symbol
+    (x + sqrt(2))*(x**2 - 2)
+    >>> factor(p, extension=True)
+    (x - sqrt(2))*(x + sqrt(2))**2
+    >>> factor(x**2 - 2, extension=True)  # all rational coefficients
+    x**2 - 2
+
+    It is also possible to use :ref:`QQ(a)` with the :py:func:`~.cancel`
+    and :py:func:`~.gcd` functions.
+
+    >>> from sympy import cancel, gcd
+    >>> cancel((x**2 - 2)/(x - sqrt(2)))
+    (x**2 - 2)/(x - sqrt(2))
+    >>> cancel((x**2 - 2)/(x - sqrt(2)), extension=sqrt(2))
+    x + sqrt(2)
+    >>> gcd(x**2 - 2, x - sqrt(2))
+    1
+    >>> gcd(x**2 - 2, x - sqrt(2), extension=sqrt(2))
+    x - sqrt(2)
+
+    When using the domain directly :ref:`QQ(a)` can be used as a constructor
+    to create instances which then support the operations ``+,-,*,**,/``. The
+    :py:meth:`~.Domain.algebraic_field` method is used to construct a
+    particular :ref:`QQ(a)` domain. The :py:meth:`~.Domain.from_sympy` method
+    can be used to create domain elements from normal SymPy expressions.
+
+    >>> K = QQ.algebraic_field(sqrt(2))
+    >>> K
+    QQ<sqrt(2)>
+    >>> xk = K.from_sympy(3 + 4*sqrt(2))
+    >>> xk  # doctest: +SKIP
+    ANP([4, 3], [1, 0, -2], QQ)
+
+    Elements of :ref:`QQ(a)` are instances of :py:class:`~.ANP` which have
+    limited printing support. The raw display shows the internal
+    representation of the element as the list ``[4, 3]`` representing the
+    coefficients of ``1`` and ``sqrt(2)`` for this element in the form
+    ``a * sqrt(2) + b * 1`` where ``a`` and ``b`` are elements of :ref:`QQ`.
+    The minimal polynomial for the generator ``(x**2 - 2)`` is also shown in
+    the :ref:`dup-representation` as the list ``[1, 0, -2]``. We can use
+    :py:meth:`~.Domain.to_sympy` to get a better printed form for the
+    elements and to see the results of operations.
+
+    >>> xk = K.from_sympy(3 + 4*sqrt(2))
+    >>> yk = K.from_sympy(2 + 3*sqrt(2))
+    >>> xk * yk  # doctest: +SKIP
+    ANP([17, 30], [1, 0, -2], QQ)
+    >>> K.to_sympy(xk * yk)
+    17*sqrt(2) + 30
+    >>> K.to_sympy(xk + yk)
+    5 + 7*sqrt(2)
+    >>> K.to_sympy(xk ** 2)
+    24*sqrt(2) + 41
+    >>> K.to_sympy(xk / yk)
+    sqrt(2)/14 + 9/7
+
+    Any expression representing an algebraic number can be used to generate
+    a :ref:`QQ(a)` domain provided its `minimal polynomial`_ can be computed.
+    The function :py:func:`~.minpoly` function is used for this.
+
+    >>> from sympy import exp, I, pi, minpoly
+    >>> g = exp(2*I*pi/3)
+    >>> g
+    exp(2*I*pi/3)
+    >>> g.is_algebraic
+    True
+    >>> minpoly(g, x)
+    x**2 + x + 1
+    >>> factor(x**3 - 1, extension=g)
+    (x - 1)*(x - exp(2*I*pi/3))*(x + 1 + exp(2*I*pi/3))
+
+    It is also possible to make an algebraic field from multiple extension
+    elements.
+
+    >>> K = QQ.algebraic_field(sqrt(2), sqrt(3))
+    >>> K
+    QQ<sqrt(2) + sqrt(3)>
+    >>> p = x**4 - 5*x**2 + 6
+    >>> factor(p)
+    (x**2 - 3)*(x**2 - 2)
+    >>> factor(p, domain=K)
+    (x - sqrt(2))*(x + sqrt(2))*(x - sqrt(3))*(x + sqrt(3))
+    >>> factor(p, extension=[sqrt(2), sqrt(3)])
+    (x - sqrt(2))*(x + sqrt(2))*(x - sqrt(3))*(x + sqrt(3))
+
+    Multiple extension elements are always combined together to make a single
+    `primitive element`_. In the case of ``[sqrt(2), sqrt(3)]`` the primitive
+    element chosen is ``sqrt(2) + sqrt(3)`` which is why the domain displays
+    as ``QQ<sqrt(2) + sqrt(3)>``. The minimal polynomial for the primitive
+    element is computed using the :py:func:`~.primitive_element` function.
+
+    >>> from sympy import primitive_element
+    >>> primitive_element([sqrt(2), sqrt(3)], x)
+    (x**4 - 10*x**2 + 1, [1, 1])
+    >>> minpoly(sqrt(2) + sqrt(3), x)
+    x**4 - 10*x**2 + 1
+
+    The extension elements that generate the domain can be accessed from the
+    domain using the :py:attr:`~.ext` and :py:attr:`~.orig_ext` attributes as
+    instances of :py:class:`~.AlgebraicNumber`. The minimal polynomial for
+    the primitive element as a :py:class:`~.DMP` instance is available as
+    :py:attr:`~.mod`.
+
+    >>> K = QQ.algebraic_field(sqrt(2), sqrt(3))
+    >>> K
+    QQ<sqrt(2) + sqrt(3)>
+    >>> K.ext
+    sqrt(2) + sqrt(3)
+    >>> K.orig_ext
+    (sqrt(2), sqrt(3))
+    >>> K.mod  # doctest: +SKIP
+    DMP_Python([1, 0, -10, 0, 1], QQ)
+
+    The `discriminant`_ of the field can be obtained from the
+    :py:meth:`~.discriminant` method, and an `integral basis`_ from the
+    :py:meth:`~.integral_basis` method. The latter returns a list of
+    :py:class:`~.ANP` instances by default, but can be made to return instances
+    of :py:class:`~.Expr` or :py:class:`~.AlgebraicNumber` by passing a ``fmt``
+    argument. The maximal order, or ring of integers, of the field can also be
+    obtained from the :py:meth:`~.maximal_order` method, as a
+    :py:class:`~sympy.polys.numberfields.modules.Submodule`.
+
+    >>> zeta5 = exp(2*I*pi/5)
+    >>> K = QQ.algebraic_field(zeta5)
+    >>> K
+    QQ<exp(2*I*pi/5)>
+    >>> K.discriminant()
+    125
+    >>> K = QQ.algebraic_field(sqrt(5))
+    >>> K
+    QQ<sqrt(5)>
+    >>> K.integral_basis(fmt='sympy')
+    [1, 1/2 + sqrt(5)/2]
+    >>> K.maximal_order()
+    Submodule[[2, 0], [1, 1]]/2
+
+    The factorization of a rational prime into prime ideals of the field is
+    computed by the :py:meth:`~.primes_above` method, which returns a list
+    of :py:class:`~sympy.polys.numberfields.primes.PrimeIdeal` instances.
+
+    >>> zeta7 = exp(2*I*pi/7)
+    >>> K = QQ.algebraic_field(zeta7)
+    >>> K
+    QQ<exp(2*I*pi/7)>
+    >>> K.primes_above(11)
+    [(11, _x**3 + 5*_x**2 + 4*_x - 1), (11, _x**3 - 4*_x**2 - 5*_x - 1)]
+
+    The Galois group of the Galois closure of the field can be computed (when
+    the minimal polynomial of the field is of sufficiently small degree).
+
+    >>> K.galois_group(by_name=True)[0]
+    S6TransitiveSubgroups.C6
+
+    Notes
+    =====
+
+    It is not currently possible to generate an algebraic extension over any
+    domain other than :ref:`QQ`. Ideally it would be possible to generate
+    extensions like ``QQ(x)(sqrt(x**2 - 2))``. This is equivalent to the
+    quotient ring ``QQ(x)[y]/(y**2 - x**2 + 2)`` and there are two
+    implementations of this kind of quotient ring/extension in the
+    :py:class:`~.QuotientRing` and :py:class:`~.MonogenicFiniteExtension`
+    classes.  Each of those implementations needs some work to make them fully
+    usable though.
+
+    .. _algebraic number field: https://en.wikipedia.org/wiki/Algebraic_number_field
+    .. _algebraic numbers: https://en.wikipedia.org/wiki/Algebraic_number
+    .. _discriminant: https://en.wikipedia.org/wiki/Discriminant_of_an_algebraic_number_field
+    .. _integral basis: https://en.wikipedia.org/wiki/Algebraic_number_field#Integral_basis
+    .. _minimal polynomial: https://en.wikipedia.org/wiki/Minimal_polynomial_(field_theory)
+    .. _primitive element: https://en.wikipedia.org/wiki/Primitive_element_theorem
+    """
+
+    dtype = ANP
+
+    is_AlgebraicField = is_Algebraic = True
+    is_Numerical = True
+
+    has_assoc_Ring = False
+    has_assoc_Field = True
+
+    def __init__(self, dom, *ext, alias=None):
+        r"""
+        Parameters
+        ==========
+
+        dom : :py:class:`~.Domain`
+            The base field over which this is an extension field.
+            Currently only :ref:`QQ` is accepted.
+
+        *ext : One or more :py:class:`~.Expr`
+            Generators of the extension. These should be expressions that are
+            algebraic over `\mathbb{Q}`.
+
+        alias : str, :py:class:`~.Symbol`, None, optional (default=None)
+            If provided, this will be used as the alias symbol for the
+            primitive element of the :py:class:`~.AlgebraicField`.
+            If ``None``, while ``ext`` consists of exactly one
+            :py:class:`~.AlgebraicNumber`, its alias (if any) will be used.
+        """
+        if not dom.is_QQ:
+            raise DomainError("ground domain must be a rational field")
+
+        from sympy.polys.numberfields import to_number_field
+        if len(ext) == 1 and isinstance(ext[0], tuple):
+            orig_ext = ext[0][1:]
+        else:
+            orig_ext = ext
+
+        if alias is None and len(ext) == 1:
+            alias = getattr(ext[0], 'alias', None)
+
+        self.orig_ext = orig_ext
+        """
+        Original elements given to generate the extension.
+
+        >>> from sympy import QQ, sqrt
+        >>> K = QQ.algebraic_field(sqrt(2), sqrt(3))
+        >>> K.orig_ext
+        (sqrt(2), sqrt(3))
+        """
+
+        self.ext = to_number_field(ext, alias=alias)
+        """
+        Primitive element used for the extension.
+
+        >>> from sympy import QQ, sqrt
+        >>> K = QQ.algebraic_field(sqrt(2), sqrt(3))
+        >>> K.ext
+        sqrt(2) + sqrt(3)
+        """
+
+        self.mod = self.ext.minpoly.rep
+        """
+        Minimal polynomial for the primitive element of the extension.
+
+        >>> from sympy import QQ, sqrt
+        >>> K = QQ.algebraic_field(sqrt(2))
+        >>> K.mod
+        DMP([1, 0, -2], QQ)
+        """
+
+        self.domain = self.dom = dom
+
+        self.ngens = 1
+        self.symbols = self.gens = (self.ext,)
+        self.unit = self([dom(1), dom(0)])
+
+        self.zero = self.dtype.zero(self.mod.to_list(), dom)
+        self.one = self.dtype.one(self.mod.to_list(), dom)
+
+        self._maximal_order = None
+        self._discriminant = None
+        self._nilradicals_mod_p = {}
+
+    def new(self, element):
+        return self.dtype(element, self.mod.to_list(), self.dom)
+
+    def __str__(self):
+        return str(self.dom) + '<' + str(self.ext) + '>'
+
+    def __hash__(self):
+        return hash((self.__class__.__name__, self.dtype, self.dom, self.ext))
+
+    def __eq__(self, other):
+        """Returns ``True`` if two domains are equivalent. """
+        if isinstance(other, AlgebraicField):
+            return self.dtype == other.dtype and self.ext == other.ext
+        else:
+            return NotImplemented
+
+    def algebraic_field(self, *extension, alias=None):
+        r"""Returns an algebraic field, i.e. `\mathbb{Q}(\alpha, \ldots)`. """
+        return AlgebraicField(self.dom, *((self.ext,) + extension), alias=alias)
+
+    def to_alg_num(self, a):
+        """Convert ``a`` of ``dtype`` to an :py:class:`~.AlgebraicNumber`. """
+        return self.ext.field_element(a)
+
+    def to_sympy(self, a):
+        """Convert ``a`` of ``dtype`` to a SymPy object. """
+        # Precompute a converter to be reused:
+        if not hasattr(self, '_converter'):
+            self._converter = _make_converter(self)
+
+        return self._converter(a)
+
+    def from_sympy(self, a):
+        """Convert SymPy's expression to ``dtype``. """
+        try:
+            return self([self.dom.from_sympy(a)])
+        except CoercionFailed:
+            pass
+
+        from sympy.polys.numberfields import to_number_field
+
+        try:
+            return self(to_number_field(a, self.ext).native_coeffs())
+        except (NotAlgebraic, IsomorphismFailed):
+            raise CoercionFailed(
+                "%s is not a valid algebraic number in %s" % (a, self))
+
+    def from_ZZ(K1, a, K0):
+        """Convert a Python ``int`` object to ``dtype``. """
+        return K1(K1.dom.convert(a, K0))
+
+    def from_ZZ_python(K1, a, K0):
+        """Convert a Python ``int`` object to ``dtype``. """
+        return K1(K1.dom.convert(a, K0))
+
+    def from_QQ(K1, a, K0):
+        """Convert a Python ``Fraction`` object to ``dtype``. """
+        return K1(K1.dom.convert(a, K0))
+
+    def from_QQ_python(K1, a, K0):
+        """Convert a Python ``Fraction`` object to ``dtype``. """
+        return K1(K1.dom.convert(a, K0))
+
+    def from_ZZ_gmpy(K1, a, K0):
+        """Convert a GMPY ``mpz`` object to ``dtype``. """
+        return K1(K1.dom.convert(a, K0))
+
+    def from_QQ_gmpy(K1, a, K0):
+        """Convert a GMPY ``mpq`` object to ``dtype``. """
+        return K1(K1.dom.convert(a, K0))
+
+    def from_RealField(K1, a, K0):
+        """Convert a mpmath ``mpf`` object to ``dtype``. """
+        return K1(K1.dom.convert(a, K0))
+
+    def get_ring(self):
+        """Returns a ring associated with ``self``. """
+        raise DomainError('there is no ring associated with %s' % self)
+
+    def is_positive(self, a):
+        """Returns True if ``a`` is positive. """
+        return self.dom.is_positive(a.LC())
+
+    def is_negative(self, a):
+        """Returns True if ``a`` is negative. """
+        return self.dom.is_negative(a.LC())
+
+    def is_nonpositive(self, a):
+        """Returns True if ``a`` is non-positive. """
+        return self.dom.is_nonpositive(a.LC())
+
+    def is_nonnegative(self, a):
+        """Returns True if ``a`` is non-negative. """
+        return self.dom.is_nonnegative(a.LC())
+
+    def numer(self, a):
+        """Returns numerator of ``a``. """
+        return a
+
+    def denom(self, a):
+        """Returns denominator of ``a``. """
+        return self.one
+
+    def from_AlgebraicField(K1, a, K0):
+        """Convert AlgebraicField element 'a' to another AlgebraicField """
+        return K1.from_sympy(K0.to_sympy(a))
+
+    def from_GaussianIntegerRing(K1, a, K0):
+        """Convert a GaussianInteger element 'a' to ``dtype``. """
+        return K1.from_sympy(K0.to_sympy(a))
+
+    def from_GaussianRationalField(K1, a, K0):
+        """Convert a GaussianRational element 'a' to ``dtype``. """
+        return K1.from_sympy(K0.to_sympy(a))
+
+    def _do_round_two(self):
+        from sympy.polys.numberfields.basis import round_two
+        ZK, dK = round_two(self, radicals=self._nilradicals_mod_p)
+        self._maximal_order = ZK
+        self._discriminant = dK
+
+    def maximal_order(self):
+        """
+        Compute the maximal order, or ring of integers, of the field.
+
+        Returns
+        =======
+
+        :py:class:`~sympy.polys.numberfields.modules.Submodule`.
+
+        See Also
+        ========
+
+        integral_basis
+
+        """
+        if self._maximal_order is None:
+            self._do_round_two()
+        return self._maximal_order
+
+    def integral_basis(self, fmt=None):
+        r"""
+        Get an integral basis for the field.
+
+        Parameters
+        ==========
+
+        fmt : str, None, optional (default=None)
+            If ``None``, return a list of :py:class:`~.ANP` instances.
+            If ``"sympy"``, convert each element of the list to an
+            :py:class:`~.Expr`, using ``self.to_sympy()``.
+            If ``"alg"``, convert each element of the list to an
+            :py:class:`~.AlgebraicNumber`, using ``self.to_alg_num()``.
+
+        Examples
+        ========
+
+        >>> from sympy import QQ, AlgebraicNumber, sqrt
+        >>> alpha = AlgebraicNumber(sqrt(5), alias='alpha')
+        >>> k = QQ.algebraic_field(alpha)
+        >>> B0 = k.integral_basis()
+        >>> B1 = k.integral_basis(fmt='sympy')
+        >>> B2 = k.integral_basis(fmt='alg')
+        >>> print(B0[1])  # doctest: +SKIP
+        ANP([mpq(1,2), mpq(1,2)], [mpq(1,1), mpq(0,1), mpq(-5,1)], QQ)
+        >>> print(B1[1])
+        1/2 + alpha/2
+        >>> print(B2[1])
+        alpha/2 + 1/2
+
+        In the last two cases we get legible expressions, which print somewhat
+        differently because of the different types involved:
+
+        >>> print(type(B1[1]))
+        <class 'sympy.core.add.Add'>
+        >>> print(type(B2[1]))
+        <class 'sympy.core.numbers.AlgebraicNumber'>
+
+        See Also
+        ========
+
+        to_sympy
+        to_alg_num
+        maximal_order
+        """
+        ZK = self.maximal_order()
+        M = ZK.QQ_matrix
+        n = M.shape[1]
+        B = [self.new(list(reversed(M[:, j].flat()))) for j in range(n)]
+        if fmt == 'sympy':
+            return [self.to_sympy(b) for b in B]
+        elif fmt == 'alg':
+            return [self.to_alg_num(b) for b in B]
+        return B
+
+    def discriminant(self):
+        """Get the discriminant of the field."""
+        if self._discriminant is None:
+            self._do_round_two()
+        return self._discriminant
+
+    def primes_above(self, p):
+        """Compute the prime ideals lying above a given rational prime *p*."""
+        from sympy.polys.numberfields.primes import prime_decomp
+        ZK = self.maximal_order()
+        dK = self.discriminant()
+        rad = self._nilradicals_mod_p.get(p)
+        return prime_decomp(p, ZK=ZK, dK=dK, radical=rad)
+
+    def galois_group(self, by_name=False, max_tries=30, randomize=False):
+        """
+        Compute the Galois group of the Galois closure of this field.
+
+        Examples
+        ========
+
+        If the field is Galois, the order of the group will equal the degree
+        of the field:
+
+        >>> from sympy import QQ
+        >>> from sympy.abc import x
+        >>> k = QQ.alg_field_from_poly(x**4 + 1)
+        >>> G, _ = k.galois_group()
+        >>> G.order()
+        4
+
+        If the field is not Galois, then its Galois closure is a proper
+        extension, and the order of the Galois group will be greater than the
+        degree of the field:
+
+        >>> k = QQ.alg_field_from_poly(x**4 - 2)
+        >>> G, _ = k.galois_group()
+        >>> G.order()
+        8
+
+        See Also
+        ========
+
+        sympy.polys.numberfields.galoisgroups.galois_group
+
+        """
+        return self.ext.minpoly_of_element().galois_group(
+            by_name=by_name, max_tries=max_tries, randomize=randomize)
+
+
+def _make_converter(K):
+    """Construct the converter to convert back to Expr"""
+    # Precompute the effect of converting to SymPy and expanding expressions
+    # like (sqrt(2) + sqrt(3))**2. Asking Expr to do the expansion on every
+    # conversion from K to Expr is slow. Here we compute the expansions for
+    # each power of the generator and collect together the resulting algebraic
+    # terms and the rational coefficients into a matrix.
+
+    ext = K.ext.as_expr()
+    todom = K.dom.from_sympy
+    toexpr = K.dom.to_sympy
+
+    if not ext.is_Add:
+        powers = [ext**n for n in range(K.mod.degree())]
+    else:
+        # primitive_element generates a QQ-linear combination of lower degree
+        # algebraic numbers to generate the higher degree extension e.g.
+        # QQ<sqrt(2)+sqrt(3)> That means that we end up having high powers of low
+        # degree algebraic numbers that can be reduced. Here we will use the
+        # minimal polynomials of the algebraic numbers to reduce those powers
+        # before converting to Expr.
+        from sympy.polys.numberfields.minpoly import minpoly
+
+        # Decompose ext as a linear combination of gens and make a symbol for
+        # each gen.
+        gens, coeffs = zip(*ext.as_coefficients_dict().items())
+        syms = symbols(f'a:{len(gens)}', cls=Dummy)
+        sym2gen = dict(zip(syms, gens))
+
+        # Make a polynomial ring that can express ext and minpolys of all gens
+        # in terms of syms.
+        R = K.dom[syms]
+        monoms = [R.ring.monomial_basis(i) for i in range(R.ngens)]
+        ext_dict = {m: todom(c) for m, c in zip(monoms, coeffs)}
+        ext_poly = R.ring.from_dict(ext_dict)
+        minpolys = [R.from_sympy(minpoly(g, s)) for s, g in sym2gen.items()]
+
+        # Compute all powers of ext_poly reduced modulo minpolys
+        powers = [R.one, ext_poly]
+        for n in range(2, K.mod.degree()):
+            ext_poly_n = (powers[-1] * ext_poly).rem(minpolys)
+            powers.append(ext_poly_n)
+
+        # Convert the powers back to Expr. This will recombine some things like
+        # sqrt(2)*sqrt(3) -> sqrt(6).
+        powers = [p.as_expr().xreplace(sym2gen) for p in powers]
+
+    # This also expands some rational powers
+    powers = [p.expand() for p in powers]
+
+    # Collect the rational coefficients and algebraic Expr that can
+    # map the ANP coefficients into an expanded SymPy expression
+    terms = [dict(t.as_coeff_Mul()[::-1] for t in Add.make_args(p)) for p in powers]
+    algebraics = set().union(*terms)
+    matrix = [[todom(t.get(a, S.Zero)) for t in terms] for a in algebraics]
+
+    # Create a function to do the conversion efficiently:
+
+    def converter(a):
+        """Convert a to Expr using converter"""
+        ai = a.to_list()[::-1]
+        coeffs_dom = [sum(mij*aj for mij, aj in zip(mi, ai)) for mi in matrix]
+        coeffs_sympy = [toexpr(c) for c in coeffs_dom]
+        res = Add(*(Mul(c, a) for c, a in zip(coeffs_sympy, algebraics)))
+        return res
+
+    return converter

.venv/lib/python3.13/site-packages/sympy/polys/domains/characteristiczero.py

@@ -0,0 +1,15 @@
+"""Implementation of :class:`CharacteristicZero` class. """
+
+
+from sympy.polys.domains.domain import Domain
+from sympy.utilities import public
+
+@public
+class CharacteristicZero(Domain):
+    """Domain that has infinite number of elements. """
+
+    has_CharacteristicZero = True
+
+    def characteristic(self):
+        """Return the characteristic of this domain. """
+        return 0

.venv/lib/python3.13/site-packages/sympy/polys/domains/complexfield.py

@@ -0,0 +1,198 @@
+"""Implementation of :class:`ComplexField` class. """
+
+
+from sympy.external.gmpy import SYMPY_INTS
+from sympy.core.numbers import Float, I
+from sympy.polys.domains.characteristiczero import CharacteristicZero
+from sympy.polys.domains.field import Field
+from sympy.polys.domains.gaussiandomains import QQ_I
+from sympy.polys.domains.simpledomain import SimpleDomain
+from sympy.polys.polyerrors import DomainError, CoercionFailed
+from sympy.utilities import public
+
+from mpmath import MPContext
+
+
+@public
+class ComplexField(Field, CharacteristicZero, SimpleDomain):
+    """Complex numbers up to the given precision. """
+
+    rep = 'CC'
+
+    is_ComplexField = is_CC = True
+
+    is_Exact = False
+    is_Numerical = True
+
+    has_assoc_Ring = False
+    has_assoc_Field = True
+
+    _default_precision = 53
+
+    @property
+    def has_default_precision(self):
+        return self.precision == self._default_precision
+
+    @property
+    def precision(self):
+        return self._context.prec
+
+    @property
+    def dps(self):
+        return self._context.dps
+
+    @property
+    def tolerance(self):
+        return self._tolerance
+
+    def __init__(self, prec=None, dps=None, tol=None):
+        # XXX: The tolerance parameter is ignored but is kept for backward
+        # compatibility for now.
+
+        context = MPContext()
+
+        if prec is None and dps is None:
+            context.prec = self._default_precision
+        elif dps is None:
+            context.prec = prec
+        elif prec is None:
+            context.dps = dps
+        else:
+            raise TypeError("Cannot set both prec and dps")
+
+        self._context = context
+
+        self._dtype = context.mpc
+        self.zero = self.dtype(0)
+        self.one = self.dtype(1)
+
+        # XXX: Neither of these is actually used anywhere.
+        self._max_denom = max(2**context.prec // 200, 99)
+        self._tolerance = self.one / self._max_denom
+
+    @property
+    def tp(self):
+        # XXX: Domain treats tp as an alias of dtype. Here we need two separate
+        # things: dtype is a callable to make/convert instances. We use tp with
+        # isinstance to check if an object is an instance of the domain
+        # already.
+        return self._dtype
+
+    def dtype(self, x, y=0):
+        # XXX: This is needed because mpmath does not recognise fmpz.
+        # It might be better to add conversion routines to mpmath and if that
+        # happens then this can be removed.
+        if isinstance(x, SYMPY_INTS):
+            x = int(x)
+        if isinstance(y, SYMPY_INTS):
+            y = int(y)
+        return self._dtype(x, y)
+
+    def __eq__(self, other):
+        return isinstance(other, ComplexField) and self.precision == other.precision
+
+    def __hash__(self):
+        return hash((self.__class__.__name__, self._dtype, self.precision))
+
+    def to_sympy(self, element):
+        """Convert ``element`` to SymPy number. """
+        return Float(element.real, self.dps) + I*Float(element.imag, self.dps)
+
+    def from_sympy(self, expr):
+        """Convert SymPy's number to ``dtype``. """
+        number = expr.evalf(n=self.dps)
+        real, imag = number.as_real_imag()
+
+        if real.is_Number and imag.is_Number:
+            return self.dtype(real, imag)
+        else:
+            raise CoercionFailed("expected complex number, got %s" % expr)
+
+    def from_ZZ(self, element, base):
+        return self.dtype(element)
+
+    def from_ZZ_gmpy(self, element, base):
+        return self.dtype(int(element))
+
+    def from_ZZ_python(self, element, base):
+        return self.dtype(element)
+
+    def from_QQ(self, element, base):
+        return self.dtype(int(element.numerator)) / int(element.denominator)
+
+    def from_QQ_python(self, element, base):
+        return self.dtype(element.numerator) / element.denominator
+
+    def from_QQ_gmpy(self, element, base):
+        return self.dtype(int(element.numerator)) / int(element.denominator)
+
+    def from_GaussianIntegerRing(self, element, base):
+        return self.dtype(int(element.x), int(element.y))
+
+    def from_GaussianRationalField(self, element, base):
+        x = element.x
+        y = element.y
+        return (self.dtype(int(x.numerator)) / int(x.denominator) +
+                self.dtype(0, int(y.numerator)) / int(y.denominator))
+
+    def from_AlgebraicField(self, element, base):
+        return self.from_sympy(base.to_sympy(element).evalf(self.dps))
+
+    def from_RealField(self, element, base):
+        return self.dtype(element)
+
+    def from_ComplexField(self, element, base):
+        return self.dtype(element)
+
+    def get_ring(self):
+        """Returns a ring associated with ``self``. """
+        raise DomainError("there is no ring associated with %s" % self)
+
+    def get_exact(self):
+        """Returns an exact domain associated with ``self``. """
+        return QQ_I
+
+    def is_negative(self, element):
+        """Returns ``False`` for any ``ComplexElement``. """
+        return False
+
+    def is_positive(self, element):
+        """Returns ``False`` for any ``ComplexElement``. """
+        return False
+
+    def is_nonnegative(self, element):
+        """Returns ``False`` for any ``ComplexElement``. """
+        return False
+
+    def is_nonpositive(self, element):
+        """Returns ``False`` for any ``ComplexElement``. """
+        return False
+
+    def gcd(self, a, b):
+        """Returns GCD of ``a`` and ``b``. """
+        return self.one
+
+    def lcm(self, a, b):
+        """Returns LCM of ``a`` and ``b``. """
+        return a*b
+
+    def almosteq(self, a, b, tolerance=None):
+        """Check if ``a`` and ``b`` are almost equal. """
+        return self._context.almosteq(a, b, tolerance)
+
+    def is_square(self, a):
+        """Returns ``True``. Every complex number has a complex square root."""
+        return True
+
+    def exsqrt(self, a):
+        r"""Returns the principal complex square root of ``a``.
+
+        Explanation
+        ===========
+        The argument of the principal square root is always within
+        $(-\frac{\pi}{2}, \frac{\pi}{2}]$. The square root may be
+        slightly inaccurate due to floating point rounding error.
+        """
+        return a ** 0.5
+
+CC = ComplexField()

.venv/lib/python3.13/site-packages/sympy/polys/domains/compositedomain.py

@@ -0,0 +1,52 @@
+"""Implementation of :class:`CompositeDomain` class. """
+
+
+from sympy.polys.domains.domain import Domain
+from sympy.polys.polyerrors import GeneratorsError
+
+from sympy.utilities import public
+
+@public
+class CompositeDomain(Domain):
+    """Base class for composite domains, e.g. ZZ[x], ZZ(X). """
+
+    is_Composite = True
+
+    gens, ngens, symbols, domain = [None]*4
+
+    def inject(self, *symbols):
+        """Inject generators into this domain.  """
+        if not (set(self.symbols) & set(symbols)):
+            return self.__class__(self.domain, self.symbols + symbols, self.order)
+        else:
+            raise GeneratorsError("common generators in %s and %s" % (self.symbols, symbols))
+
+    def drop(self, *symbols):
+        """Drop generators from this domain. """
+        symset = set(symbols)
+        newsyms = tuple(s for s in self.symbols if s not in symset)
+        domain = self.domain.drop(*symbols)
+        if not newsyms:
+            return domain
+        else:
+            return self.__class__(domain, newsyms, self.order)
+
+    def set_domain(self, domain):
+        """Set the ground domain of this domain. """
+        return self.__class__(domain, self.symbols, self.order)
+
+    @property
+    def is_Exact(self):
+        """Returns ``True`` if this domain is exact. """
+        return self.domain.is_Exact
+
+    def get_exact(self):
+        """Returns an exact version of this domain. """
+        return self.set_domain(self.domain.get_exact())
+
+    @property
+    def has_CharacteristicZero(self):
+        return self.domain.has_CharacteristicZero
+
+    def characteristic(self):
+        return self.domain.characteristic()

.venv/lib/python3.13/site-packages/sympy/polys/domains/domain.py

@@ -0,0 +1,1382 @@
+"""Implementation of :class:`Domain` class. """
+
+from __future__ import annotations
+from typing import Any
+
+from sympy.core.numbers import AlgebraicNumber
+from sympy.core import Basic, sympify
+from sympy.core.sorting import ordered
+from sympy.external.gmpy import GROUND_TYPES
+from sympy.polys.domains.domainelement import DomainElement
+from sympy.polys.orderings import lex
+from sympy.polys.polyerrors import UnificationFailed, CoercionFailed, DomainError
+from sympy.polys.polyutils import _unify_gens, _not_a_coeff
+from sympy.utilities import public
+from sympy.utilities.iterables import is_sequence
+
+
+@public
+class Domain:
+    """Superclass for all domains in the polys domains system.
+
+    See :ref:`polys-domainsintro` for an introductory explanation of the
+    domains system.
+
+    The :py:class:`~.Domain` class is an abstract base class for all of the
+    concrete domain types. There are many different :py:class:`~.Domain`
+    subclasses each of which has an associated ``dtype`` which is a class
+    representing the elements of the domain. The coefficients of a
+    :py:class:`~.Poly` are elements of a domain which must be a subclass of
+    :py:class:`~.Domain`.
+
+    Examples
+    ========
+
+    The most common example domains are the integers :ref:`ZZ` and the
+    rationals :ref:`QQ`.
+
+    >>> from sympy import Poly, symbols, Domain
+    >>> x, y = symbols('x, y')
+    >>> p = Poly(x**2 + y)
+    >>> p
+    Poly(x**2 + y, x, y, domain='ZZ')
+    >>> p.domain
+    ZZ
+    >>> isinstance(p.domain, Domain)
+    True
+    >>> Poly(x**2 + y/2)
+    Poly(x**2 + 1/2*y, x, y, domain='QQ')
+
+    The domains can be used directly in which case the domain object e.g.
+    (:ref:`ZZ` or :ref:`QQ`) can be used as a constructor for elements of
+    ``dtype``.
+
+    >>> from sympy import ZZ, QQ
+    >>> ZZ(2)
+    2
+    >>> ZZ.dtype  # doctest: +SKIP
+    <class 'int'>
+    >>> type(ZZ(2))  # doctest: +SKIP
+    <class 'int'>
+    >>> QQ(1, 2)
+    1/2
+    >>> type(QQ(1, 2))  # doctest: +SKIP
+    <class 'sympy.polys.domains.pythonrational.PythonRational'>
+
+    The corresponding domain elements can be used with the arithmetic
+    operations ``+,-,*,**`` and depending on the domain some combination of
+    ``/,//,%`` might be usable. For example in :ref:`ZZ` both ``//`` (floor
+    division) and ``%`` (modulo division) can be used but ``/`` (true
+    division) cannot. Since :ref:`QQ` is a :py:class:`~.Field` its elements
+    can be used with ``/`` but ``//`` and ``%`` should not be used. Some
+    domains have a :py:meth:`~.Domain.gcd` method.
+
+    >>> ZZ(2) + ZZ(3)
+    5
+    >>> ZZ(5) // ZZ(2)
+    2
+    >>> ZZ(5) % ZZ(2)
+    1
+    >>> QQ(1, 2) / QQ(2, 3)
+    3/4
+    >>> ZZ.gcd(ZZ(4), ZZ(2))
+    2
+    >>> QQ.gcd(QQ(2,7), QQ(5,3))
+    1/21
+    >>> ZZ.is_Field
+    False
+    >>> QQ.is_Field
+    True
+
+    There are also many other domains including:
+
+        1. :ref:`GF(p)` for finite fields of prime order.
+        2. :ref:`RR` for real (floating point) numbers.
+        3. :ref:`CC` for complex (floating point) numbers.
+        4. :ref:`QQ(a)` for algebraic number fields.
+        5. :ref:`K[x]` for polynomial rings.
+        6. :ref:`K(x)` for rational function fields.
+        7. :ref:`EX` for arbitrary expressions.
+
+    Each domain is represented by a domain object and also an implementation
+    class (``dtype``) for the elements of the domain. For example the
+    :ref:`K[x]` domains are represented by a domain object which is an
+    instance of :py:class:`~.PolynomialRing` and the elements are always
+    instances of :py:class:`~.PolyElement`. The implementation class
+    represents particular types of mathematical expressions in a way that is
+    more efficient than a normal SymPy expression which is of type
+    :py:class:`~.Expr`. The domain methods :py:meth:`~.Domain.from_sympy` and
+    :py:meth:`~.Domain.to_sympy` are used to convert from :py:class:`~.Expr`
+    to a domain element and vice versa.
+
+    >>> from sympy import Symbol, ZZ, Expr
+    >>> x = Symbol('x')
+    >>> K = ZZ[x]           # polynomial ring domain
+    >>> K
+    ZZ[x]
+    >>> type(K)             # class of the domain
+    <class 'sympy.polys.domains.polynomialring.PolynomialRing'>
+    >>> K.dtype             # doctest: +SKIP
+    <class 'sympy.polys.rings.PolyElement'>
+    >>> p_expr = x**2 + 1   # Expr
+    >>> p_expr
+    x**2 + 1
+    >>> type(p_expr)
+    <class 'sympy.core.add.Add'>
+    >>> isinstance(p_expr, Expr)
+    True
+    >>> p_domain = K.from_sympy(p_expr)
+    >>> p_domain            # domain element
+    x**2 + 1
+    >>> type(p_domain)
+    <class 'sympy.polys.rings.PolyElement'>
+    >>> K.to_sympy(p_domain) == p_expr
+    True
+
+    The :py:meth:`~.Domain.convert_from` method is used to convert domain
+    elements from one domain to another.
+
+    >>> from sympy import ZZ, QQ
+    >>> ez = ZZ(2)
+    >>> eq = QQ.convert_from(ez, ZZ)
+    >>> type(ez)  # doctest: +SKIP
+    <class 'int'>
+    >>> type(eq)  # doctest: +SKIP
+    <class 'sympy.polys.domains.pythonrational.PythonRational'>
+
+    Elements from different domains should not be mixed in arithmetic or other
+    operations: they should be converted to a common domain first.  The domain
+    method :py:meth:`~.Domain.unify` is used to find a domain that can
+    represent all the elements of two given domains.
+
+    >>> from sympy import ZZ, QQ, symbols
+    >>> x, y = symbols('x, y')
+    >>> ZZ.unify(QQ)
+    QQ
+    >>> ZZ[x].unify(QQ)
+    QQ[x]
+    >>> ZZ[x].unify(QQ[y])
+    QQ[x,y]
+
+    If a domain is a :py:class:`~.Ring` then is might have an associated
+    :py:class:`~.Field` and vice versa. The :py:meth:`~.Domain.get_field` and
+    :py:meth:`~.Domain.get_ring` methods will find or create the associated
+    domain.
+
+    >>> from sympy import ZZ, QQ, Symbol
+    >>> x = Symbol('x')
+    >>> ZZ.has_assoc_Field
+    True
+    >>> ZZ.get_field()
+    QQ
+    >>> QQ.has_assoc_Ring
+    True
+    >>> QQ.get_ring()
+    ZZ
+    >>> K = QQ[x]
+    >>> K
+    QQ[x]
+    >>> K.get_field()
+    QQ(x)
+
+    See also
+    ========
+
+    DomainElement: abstract base class for domain elements
+    construct_domain: construct a minimal domain for some expressions
+
+    """
+
+    dtype: type | None = None
+    """The type (class) of the elements of this :py:class:`~.Domain`:
+
+    >>> from sympy import ZZ, QQ, Symbol
+    >>> ZZ.dtype
+    <class 'int'>
+    >>> z = ZZ(2)
+    >>> z
+    2
+    >>> type(z)
+    <class 'int'>
+    >>> type(z) == ZZ.dtype
+    True
+
+    Every domain has an associated **dtype** ("datatype") which is the
+    class of the associated domain elements.
+
+    See also
+    ========
+
+    of_type
+    """
+
+    zero: Any = None
+    """The zero element of the :py:class:`~.Domain`:
+
+    >>> from sympy import QQ
+    >>> QQ.zero
+    0
+    >>> QQ.of_type(QQ.zero)
+    True
+
+    See also
+    ========
+
+    of_type
+    one
+    """
+
+    one: Any = None
+    """The one element of the :py:class:`~.Domain`:
+
+    >>> from sympy import QQ
+    >>> QQ.one
+    1
+    >>> QQ.of_type(QQ.one)
+    True
+
+    See also
+    ========
+
+    of_type
+    zero
+    """
+
+    is_Ring = False
+    """Boolean flag indicating if the domain is a :py:class:`~.Ring`.
+
+    >>> from sympy import ZZ
+    >>> ZZ.is_Ring
+    True
+
+    Basically every :py:class:`~.Domain` represents a ring so this flag is
+    not that useful.
+
+    See also
+    ========
+
+    is_PID
+    is_Field
+    get_ring
+    has_assoc_Ring
+    """
+
+    is_Field = False
+    """Boolean flag indicating if the domain is a :py:class:`~.Field`.
+
+    >>> from sympy import ZZ, QQ
+    >>> ZZ.is_Field
+    False
+    >>> QQ.is_Field
+    True
+
+    See also
+    ========
+
+    is_PID
+    is_Ring
+    get_field
+    has_assoc_Field
+    """
+
+    has_assoc_Ring = False
+    """Boolean flag indicating if the domain has an associated
+    :py:class:`~.Ring`.
+
+    >>> from sympy import QQ
+    >>> QQ.has_assoc_Ring
+    True
+    >>> QQ.get_ring()
+    ZZ
+
+    See also
+    ========
+
+    is_Field
+    get_ring
+    """
+
+    has_assoc_Field = False
+    """Boolean flag indicating if the domain has an associated
+    :py:class:`~.Field`.
+
+    >>> from sympy import ZZ
+    >>> ZZ.has_assoc_Field
+    True
+    >>> ZZ.get_field()
+    QQ
+
+    See also
+    ========
+
+    is_Field
+    get_field
+    """
+
+    is_FiniteField = is_FF = False
+    is_IntegerRing = is_ZZ = False
+    is_RationalField = is_QQ = False
+    is_GaussianRing = is_ZZ_I = False
+    is_GaussianField = is_QQ_I = False
+    is_RealField = is_RR = False
+    is_ComplexField = is_CC = False
+    is_AlgebraicField = is_Algebraic = False
+    is_PolynomialRing = is_Poly = False
+    is_FractionField = is_Frac = False
+    is_SymbolicDomain = is_EX = False
+    is_SymbolicRawDomain = is_EXRAW = False
+    is_FiniteExtension = False
+
+    is_Exact = True
+    is_Numerical = False
+
+    is_Simple = False
+    is_Composite = False
+
+    is_PID = False
+    """Boolean flag indicating if the domain is a `principal ideal domain`_.
+
+    >>> from sympy import ZZ
+    >>> ZZ.has_assoc_Field
+    True
+    >>> ZZ.get_field()
+    QQ
+
+    .. _principal ideal domain: https://en.wikipedia.org/wiki/Principal_ideal_domain
+
+    See also
+    ========
+
+    is_Field
+    get_field
+    """
+
+    has_CharacteristicZero = False
+
+    rep: str | None = None
+    alias: str | None = None
+
+    def __init__(self):
+        raise NotImplementedError
+
+    def __str__(self):
+        return self.rep
+
+    def __repr__(self):
+        return str(self)
+
+    def __hash__(self):
+        return hash((self.__class__.__name__, self.dtype))
+
+    def new(self, *args):
+        return self.dtype(*args)
+
+    @property
+    def tp(self):
+        """Alias for :py:attr:`~.Domain.dtype`"""
+        return self.dtype
+
+    def __call__(self, *args):
+        """Construct an element of ``self`` domain from ``args``. """
+        return self.new(*args)
+
+    def normal(self, *args):
+        return self.dtype(*args)
+
+    def convert_from(self, element, base):
+        """Convert ``element`` to ``self.dtype`` given the base domain. """
+        if base.alias is not None:
+            method = "from_" + base.alias
+        else:
+            method = "from_" + base.__class__.__name__
+
+        _convert = getattr(self, method)
+
+        if _convert is not None:
+            result = _convert(element, base)
+
+            if result is not None:
+                return result
+
+        raise CoercionFailed("Cannot convert %s of type %s from %s to %s" % (element, type(element), base, self))
+
+    def convert(self, element, base=None):
+        """Convert ``element`` to ``self.dtype``. """
+
+        if base is not None:
+            if _not_a_coeff(element):
+                raise CoercionFailed('%s is not in any domain' % element)
+            return self.convert_from(element, base)
+
+        if self.of_type(element):
+            return element
+
+        if _not_a_coeff(element):
+            raise CoercionFailed('%s is not in any domain' % element)
+
+        from sympy.polys.domains import ZZ, QQ, RealField, ComplexField
+
+        if ZZ.of_type(element):
+            return self.convert_from(element, ZZ)
+
+        if isinstance(element, int):
+            return self.convert_from(ZZ(element), ZZ)
+
+        if GROUND_TYPES != 'python':
+            if isinstance(element, ZZ.tp):
+                return self.convert_from(element, ZZ)
+            if isinstance(element, QQ.tp):
+                return self.convert_from(element, QQ)
+
+        if isinstance(element, float):
+            parent = RealField()
+            return self.convert_from(parent(element), parent)
+
+        if isinstance(element, complex):
+            parent = ComplexField()
+            return self.convert_from(parent(element), parent)
+
+        if type(element).__name__ == 'mpf':
+            parent = RealField()
+            return self.convert_from(parent(element), parent)
+
+        if type(element).__name__ == 'mpc':
+            parent = ComplexField()
+            return self.convert_from(parent(element), parent)
+
+        if isinstance(element, DomainElement):
+            return self.convert_from(element, element.parent())
+
+        # TODO: implement this in from_ methods
+        if self.is_Numerical and getattr(element, 'is_ground', False):
+            return self.convert(element.LC())
+
+        if isinstance(element, Basic):
+            try:
+                return self.from_sympy(element)
+            except (TypeError, ValueError):
+                pass
+        else: # TODO: remove this branch
+            if not is_sequence(element):
+                try:
+                    element = sympify(element, strict=True)
+                    if isinstance(element, Basic):
+                        return self.from_sympy(element)
+                except (TypeError, ValueError):
+                    pass
+
+        raise CoercionFailed("Cannot convert %s of type %s to %s" % (element, type(element), self))
+
+    def of_type(self, element):
+        """Check if ``a`` is of type ``dtype``. """
+        return isinstance(element, self.tp)
+
+    def __contains__(self, a):
+        """Check if ``a`` belongs to this domain. """
+        try:
+            if _not_a_coeff(a):
+                raise CoercionFailed
+            self.convert(a)  # this might raise, too
+        except CoercionFailed:
+            return False
+
+        return True
+
+    def to_sympy(self, a):
+        """Convert domain element *a* to a SymPy expression (Expr).
+
+        Explanation
+        ===========
+
+        Convert a :py:class:`~.Domain` element *a* to :py:class:`~.Expr`. Most
+        public SymPy functions work with objects of type :py:class:`~.Expr`.
+        The elements of a :py:class:`~.Domain` have a different internal
+        representation. It is not possible to mix domain elements with
+        :py:class:`~.Expr` so each domain has :py:meth:`~.Domain.to_sympy` and
+        :py:meth:`~.Domain.from_sympy` methods to convert its domain elements
+        to and from :py:class:`~.Expr`.
+
+        Parameters
+        ==========
+
+        a: domain element
+            An element of this :py:class:`~.Domain`.
+
+        Returns
+        =======
+
+        expr: Expr
+            A normal SymPy expression of type :py:class:`~.Expr`.
+
+        Examples
+        ========
+
+        Construct an element of the :ref:`QQ` domain and then convert it to
+        :py:class:`~.Expr`.
+
+        >>> from sympy import QQ, Expr
+        >>> q_domain = QQ(2)
+        >>> q_domain
+        2
+        >>> q_expr = QQ.to_sympy(q_domain)
+        >>> q_expr
+        2
+
+        Although the printed forms look similar these objects are not of the
+        same type.
+
+        >>> isinstance(q_domain, Expr)
+        False
+        >>> isinstance(q_expr, Expr)
+        True
+
+        Construct an element of :ref:`K[x]` and convert to
+        :py:class:`~.Expr`.
+
+        >>> from sympy import Symbol
+        >>> x = Symbol('x')
+        >>> K = QQ[x]
+        >>> x_domain = K.gens[0]  # generator x as a domain element
+        >>> p_domain = x_domain**2/3 + 1
+        >>> p_domain
+        1/3*x**2 + 1
+        >>> p_expr = K.to_sympy(p_domain)
+        >>> p_expr
+        x**2/3 + 1
+
+        The :py:meth:`~.Domain.from_sympy` method is used for the opposite
+        conversion from a normal SymPy expression to a domain element.
+
+        >>> p_domain == p_expr
+        False
+        >>> K.from_sympy(p_expr) == p_domain
+        True
+        >>> K.to_sympy(p_domain) == p_expr
+        True
+        >>> K.from_sympy(K.to_sympy(p_domain)) == p_domain
+        True
+        >>> K.to_sympy(K.from_sympy(p_expr)) == p_expr
+        True
+
+        The :py:meth:`~.Domain.from_sympy` method makes it easier to construct
+        domain elements interactively.
+
+        >>> from sympy import Symbol
+        >>> x = Symbol('x')
+        >>> K = QQ[x]
+        >>> K.from_sympy(x**2/3 + 1)
+        1/3*x**2 + 1
+
+        See also
+        ========
+
+        from_sympy
+        convert_from
+        """
+        raise NotImplementedError
+
+    def from_sympy(self, a):
+        """Convert a SymPy expression to an element of this domain.
+
+        Explanation
+        ===========
+
+        See :py:meth:`~.Domain.to_sympy` for explanation and examples.
+
+        Parameters
+        ==========
+
+        expr: Expr
+            A normal SymPy expression of type :py:class:`~.Expr`.
+
+        Returns
+        =======
+
+        a: domain element
+            An element of this :py:class:`~.Domain`.
+
+        See also
+        ========
+
+        to_sympy
+        convert_from
+        """
+        raise NotImplementedError
+
+    def sum(self, args):
+        return sum(args, start=self.zero)
+
+    def from_FF(K1, a, K0):
+        """Convert ``ModularInteger(int)`` to ``dtype``. """
+        return None
+
+    def from_FF_python(K1, a, K0):
+        """Convert ``ModularInteger(int)`` to ``dtype``. """
+        return None
+
+    def from_ZZ_python(K1, a, K0):
+        """Convert a Python ``int`` object to ``dtype``. """
+        return None
+
+    def from_QQ_python(K1, a, K0):
+        """Convert a Python ``Fraction`` object to ``dtype``. """
+        return None
+
+    def from_FF_gmpy(K1, a, K0):
+        """Convert ``ModularInteger(mpz)`` to ``dtype``. """
+        return None
+
+    def from_ZZ_gmpy(K1, a, K0):
+        """Convert a GMPY ``mpz`` object to ``dtype``. """
+        return None
+
+    def from_QQ_gmpy(K1, a, K0):
+        """Convert a GMPY ``mpq`` object to ``dtype``. """
+        return None
+
+    def from_RealField(K1, a, K0):
+        """Convert a real element object to ``dtype``. """
+        return None
+
+    def from_ComplexField(K1, a, K0):
+        """Convert a complex element to ``dtype``. """
+        return None
+
+    def from_AlgebraicField(K1, a, K0):
+        """Convert an algebraic number to ``dtype``. """
+        return None
+
+    def from_PolynomialRing(K1, a, K0):
+        """Convert a polynomial to ``dtype``. """
+        if a.is_ground:
+            return K1.convert(a.LC, K0.dom)
+
+    def from_FractionField(K1, a, K0):
+        """Convert a rational function to ``dtype``. """
+        return None
+
+    def from_MonogenicFiniteExtension(K1, a, K0):
+        """Convert an ``ExtensionElement`` to ``dtype``. """
+        return K1.convert_from(a.rep, K0.ring)
+
+    def from_ExpressionDomain(K1, a, K0):
+        """Convert a ``EX`` object to ``dtype``. """
+        return K1.from_sympy(a.ex)
+
+    def from_ExpressionRawDomain(K1, a, K0):
+        """Convert a ``EX`` object to ``dtype``. """
+        return K1.from_sympy(a)
+
+    def from_GlobalPolynomialRing(K1, a, K0):
+        """Convert a polynomial to ``dtype``. """
+        if a.degree() <= 0:
+            return K1.convert(a.LC(), K0.dom)
+
+    def from_GeneralizedPolynomialRing(K1, a, K0):
+        return K1.from_FractionField(a, K0)
+
+    def unify_with_symbols(K0, K1, symbols):
+        if (K0.is_Composite and (set(K0.symbols) & set(symbols))) or (K1.is_Composite and (set(K1.symbols) & set(symbols))):
+            raise UnificationFailed("Cannot unify %s with %s, given %s generators" % (K0, K1, tuple(symbols)))
+
+        return K0.unify(K1)
+
+    def unify_composite(K0, K1):
+        """Unify two domains where at least one is composite."""
+        K0_ground = K0.dom if K0.is_Composite else K0
+        K1_ground = K1.dom if K1.is_Composite else K1
+
+        K0_symbols = K0.symbols if K0.is_Composite else ()
+        K1_symbols = K1.symbols if K1.is_Composite else ()
+
+        domain = K0_ground.unify(K1_ground)
+        symbols = _unify_gens(K0_symbols, K1_symbols)
+        order = K0.order if K0.is_Composite else K1.order
+
+        # E.g. ZZ[x].unify(QQ.frac_field(x)) -> ZZ.frac_field(x)
+        if ((K0.is_FractionField and K1.is_PolynomialRing or
+             K1.is_FractionField and K0.is_PolynomialRing) and
+             (not K0_ground.is_Field or not K1_ground.is_Field) and domain.is_Field
+             and domain.has_assoc_Ring):
+            domain = domain.get_ring()
+
+        if K0.is_Composite and (not K1.is_Composite or K0.is_FractionField or K1.is_PolynomialRing):
+            cls = K0.__class__
+        else:
+            cls = K1.__class__
+
+        # Here cls might be PolynomialRing, FractionField, GlobalPolynomialRing
+        # (dense/old Polynomialring) or dense/old FractionField.
+
+        from sympy.polys.domains.old_polynomialring import GlobalPolynomialRing
+        if cls == GlobalPolynomialRing:
+            return cls(domain, symbols)
+
+        return cls(domain, symbols, order)
+
+    def unify(K0, K1, symbols=None):
+        """
+        Construct a minimal domain that contains elements of ``K0`` and ``K1``.
+
+        Known domains (from smallest to largest):
+
+        - ``GF(p)``
+        - ``ZZ``
+        - ``QQ``
+        - ``RR(prec, tol)``
+        - ``CC(prec, tol)``
+        - ``ALG(a, b, c)``
+        - ``K[x, y, z]``
+        - ``K(x, y, z)``
+        - ``EX``
+
+        """
+        if symbols is not None:
+            return K0.unify_with_symbols(K1, symbols)
+
+        if K0 == K1:
+            return K0
+
+        if not (K0.has_CharacteristicZero and K1.has_CharacteristicZero):
+            # Reject unification of domains with different characteristics.
+            if K0.characteristic() != K1.characteristic():
+                raise UnificationFailed("Cannot unify %s with %s" % (K0, K1))
+
+            # We do not get here if K0 == K1. The two domains have the same
+            # characteristic but are unequal so at least one is composite and
+            # we are unifying something like GF(3).unify(GF(3)[x]).
+            return K0.unify_composite(K1)
+
+        # From here we know both domains have characteristic zero and it can be
+        # acceptable to fall back on EX.
+
+        if K0.is_EXRAW:
+            return K0
+        if K1.is_EXRAW:
+            return K1
+
+        if K0.is_EX:
+            return K0
+        if K1.is_EX:
+            return K1
+
+        if K0.is_FiniteExtension or K1.is_FiniteExtension:
+            if K1.is_FiniteExtension:
+                K0, K1 = K1, K0
+            if K1.is_FiniteExtension:
+                # Unifying two extensions.
+                # Try to ensure that K0.unify(K1) == K1.unify(K0)
+                if list(ordered([K0.modulus, K1.modulus]))[1] == K0.modulus:
+                    K0, K1 = K1, K0
+                return K1.set_domain(K0)
+            else:
+                # Drop the generator from other and unify with the base domain
+                K1 = K1.drop(K0.symbol)
+                K1 = K0.domain.unify(K1)
+                return K0.set_domain(K1)
+
+        if K0.is_Composite or K1.is_Composite:
+            return K0.unify_composite(K1)
+
+        if K1.is_ComplexField:
+            K0, K1 = K1, K0
+        if K0.is_ComplexField:
+            if K1.is_ComplexField or K1.is_RealField:
+                if K0.precision >= K1.precision:
+                    return K0
+                else:
+                    from sympy.polys.domains.complexfield import ComplexField
+                    return ComplexField(prec=K1.precision)
+            else:
+                return K0
+
+        if K1.is_RealField:
+            K0, K1 = K1, K0
+        if K0.is_RealField:
+            if K1.is_RealField:
+                if K0.precision >= K1.precision:
+                    return K0
+                else:
+                    return K1
+            elif K1.is_GaussianRing or K1.is_GaussianField:
+                from sympy.polys.domains.complexfield import ComplexField
+                return ComplexField(prec=K0.precision)
+            else:
+                return K0
+
+        if K1.is_AlgebraicField:
+            K0, K1 = K1, K0
+        if K0.is_AlgebraicField:
+            if K1.is_GaussianRing:
+                K1 = K1.get_field()
+            if K1.is_GaussianField:
+                K1 = K1.as_AlgebraicField()
+            if K1.is_AlgebraicField:
+                return K0.__class__(K0.dom.unify(K1.dom), *_unify_gens(K0.orig_ext, K1.orig_ext))
+            else:
+                return K0
+
+        if K0.is_GaussianField:
+            return K0
+        if K1.is_GaussianField:
+            return K1
+
+        if K0.is_GaussianRing:
+            if K1.is_RationalField:
+                K0 = K0.get_field()
+            return K0
+        if K1.is_GaussianRing:
+            if K0.is_RationalField:
+                K1 = K1.get_field()
+            return K1
+
+        if K0.is_RationalField:
+            return K0
+        if K1.is_RationalField:
+            return K1
+
+        if K0.is_IntegerRing:
+            return K0
+        if K1.is_IntegerRing:
+            return K1
+
+        from sympy.polys.domains import EX
+        return EX
+
+    def __eq__(self, other):
+        """Returns ``True`` if two domains are equivalent. """
+        # XXX: Remove this.
+        return isinstance(other, Domain) and self.dtype == other.dtype
+
+    def __ne__(self, other):
+        """Returns ``False`` if two domains are equivalent. """
+        return not self == other
+
+    def map(self, seq):
+        """Rersively apply ``self`` to all elements of ``seq``. """
+        result = []
+
+        for elt in seq:
+            if isinstance(elt, list):
+                result.append(self.map(elt))
+            else:
+                result.append(self(elt))
+
+        return result
+
+    def get_ring(self):
+        """Returns a ring associated with ``self``. """
+        raise DomainError('there is no ring associated with %s' % self)
+
+    def get_field(self):
+        """Returns a field associated with ``self``. """
+        raise DomainError('there is no field associated with %s' % self)
+
+    def get_exact(self):
+        """Returns an exact domain associated with ``self``. """
+        return self
+
+    def __getitem__(self, symbols):
+        """The mathematical way to make a polynomial ring. """
+        if hasattr(symbols, '__iter__'):
+            return self.poly_ring(*symbols)
+        else:
+            return self.poly_ring(symbols)
+
+    def poly_ring(self, *symbols, order=lex):
+        """Returns a polynomial ring, i.e. `K[X]`. """
+        from sympy.polys.domains.polynomialring import PolynomialRing
+        return PolynomialRing(self, symbols, order)
+
+    def frac_field(self, *symbols, order=lex):
+        """Returns a fraction field, i.e. `K(X)`. """
+        from sympy.polys.domains.fractionfield import FractionField
+        return FractionField(self, symbols, order)
+
+    def old_poly_ring(self, *symbols, **kwargs):
+        """Returns a polynomial ring, i.e. `K[X]`. """
+        from sympy.polys.domains.old_polynomialring import PolynomialRing
+        return PolynomialRing(self, *symbols, **kwargs)
+
+    def old_frac_field(self, *symbols, **kwargs):
+        """Returns a fraction field, i.e. `K(X)`. """
+        from sympy.polys.domains.old_fractionfield import FractionField
+        return FractionField(self, *symbols, **kwargs)
+
+    def algebraic_field(self, *extension, alias=None):
+        r"""Returns an algebraic field, i.e. `K(\alpha, \ldots)`. """
+        raise DomainError("Cannot create algebraic field over %s" % self)
+
+    def alg_field_from_poly(self, poly, alias=None, root_index=-1):
+        r"""
+        Convenience method to construct an algebraic extension on a root of a
+        polynomial, chosen by root index.
+
+        Parameters
+        ==========
+
+        poly : :py:class:`~.Poly`
+            The polynomial whose root generates the extension.
+        alias : str, optional (default=None)
+            Symbol name for the generator of the extension.
+            E.g. "alpha" or "theta".
+        root_index : int, optional (default=-1)
+            Specifies which root of the polynomial is desired. The ordering is
+            as defined by the :py:class:`~.ComplexRootOf` class. The default of
+            ``-1`` selects the most natural choice in the common cases of
+            quadratic and cyclotomic fields (the square root on the positive
+            real or imaginary axis, resp. $\mathrm{e}^{2\pi i/n}$).
+
+        Examples
+        ========
+
+        >>> from sympy import QQ, Poly
+        >>> from sympy.abc import x
+        >>> f = Poly(x**2 - 2)
+        >>> K = QQ.alg_field_from_poly(f)
+        >>> K.ext.minpoly == f
+        True
+        >>> g = Poly(8*x**3 - 6*x - 1)
+        >>> L = QQ.alg_field_from_poly(g, "alpha")
+        >>> L.ext.minpoly == g
+        True
+        >>> L.to_sympy(L([1, 1, 1]))
+        alpha**2 + alpha + 1
+
+        """
+        from sympy.polys.rootoftools import CRootOf
+        root = CRootOf(poly, root_index)
+        alpha = AlgebraicNumber(root, alias=alias)
+        return self.algebraic_field(alpha, alias=alias)
+
+    def cyclotomic_field(self, n, ss=False, alias="zeta", gen=None, root_index=-1):
+        r"""
+        Convenience method to construct a cyclotomic field.
+
+        Parameters
+        ==========
+
+        n : int
+            Construct the nth cyclotomic field.
+        ss : boolean, optional (default=False)
+            If True, append *n* as a subscript on the alias string.
+        alias : str, optional (default="zeta")
+            Symbol name for the generator.
+        gen : :py:class:`~.Symbol`, optional (default=None)
+            Desired variable for the cyclotomic polynomial that defines the
+            field. If ``None``, a dummy variable will be used.
+        root_index : int, optional (default=-1)
+            Specifies which root of the polynomial is desired. The ordering is
+            as defined by the :py:class:`~.ComplexRootOf` class. The default of
+            ``-1`` selects the root $\mathrm{e}^{2\pi i/n}$.
+
+        Examples
+        ========
+
+        >>> from sympy import QQ, latex
+        >>> K = QQ.cyclotomic_field(5)
+        >>> K.to_sympy(K([-1, 1]))
+        1 - zeta
+        >>> L = QQ.cyclotomic_field(7, True)
+        >>> a = L.to_sympy(L([-1, 1]))
+        >>> print(a)
+        1 - zeta7
+        >>> print(latex(a))
+        1 - \zeta_{7}
+
+        """
+        from sympy.polys.specialpolys import cyclotomic_poly
+        if ss:
+            alias += str(n)
+        return self.alg_field_from_poly(cyclotomic_poly(n, gen), alias=alias,
+                                        root_index=root_index)
+
+    def inject(self, *symbols):
+        """Inject generators into this domain. """
+        raise NotImplementedError
+
+    def drop(self, *symbols):
+        """Drop generators from this domain. """
+        if self.is_Simple:
+            return self
+        raise NotImplementedError  # pragma: no cover
+
+    def is_zero(self, a):
+        """Returns True if ``a`` is zero. """
+        return not a
+
+    def is_one(self, a):
+        """Returns True if ``a`` is one. """
+        return a == self.one
+
+    def is_positive(self, a):
+        """Returns True if ``a`` is positive. """
+        return a > 0
+
+    def is_negative(self, a):
+        """Returns True if ``a`` is negative. """
+        return a < 0
+
+    def is_nonpositive(self, a):
+        """Returns True if ``a`` is non-positive. """
+        return a <= 0
+
+    def is_nonnegative(self, a):
+        """Returns True if ``a`` is non-negative. """
+        return a >= 0
+
+    def canonical_unit(self, a):
+        if self.is_negative(a):
+            return -self.one
+        else:
+            return self.one
+
+    def abs(self, a):
+        """Absolute value of ``a``, implies ``__abs__``. """
+        return abs(a)
+
+    def neg(self, a):
+        """Returns ``a`` negated, implies ``__neg__``. """
+        return -a
+
+    def pos(self, a):
+        """Returns ``a`` positive, implies ``__pos__``. """
+        return +a
+
+    def add(self, a, b):
+        """Sum of ``a`` and ``b``, implies ``__add__``.  """
+        return a + b
+
+    def sub(self, a, b):
+        """Difference of ``a`` and ``b``, implies ``__sub__``.  """
+        return a - b
+
+    def mul(self, a, b):
+        """Product of ``a`` and ``b``, implies ``__mul__``.  """
+        return a * b
+
+    def pow(self, a, b):
+        """Raise ``a`` to power ``b``, implies ``__pow__``.  """
+        return a ** b
+
+    def exquo(self, a, b):
+        """Exact quotient of *a* and *b*. Analogue of ``a / b``.
+
+        Explanation
+        ===========
+
+        This is essentially the same as ``a / b`` except that an error will be
+        raised if the division is inexact (if there is any remainder) and the
+        result will always be a domain element. When working in a
+        :py:class:`~.Domain` that is not a :py:class:`~.Field` (e.g. :ref:`ZZ`
+        or :ref:`K[x]`) ``exquo`` should be used instead of ``/``.
+
+        The key invariant is that if ``q = K.exquo(a, b)`` (and ``exquo`` does
+        not raise an exception) then ``a == b*q``.
+
+        Examples
+        ========
+
+        We can use ``K.exquo`` instead of ``/`` for exact division.
+
+        >>> from sympy import ZZ
+        >>> ZZ.exquo(ZZ(4), ZZ(2))
+        2
+        >>> ZZ.exquo(ZZ(5), ZZ(2))
+        Traceback (most recent call last):
+            ...
+        ExactQuotientFailed: 2 does not divide 5 in ZZ
+
+        Over a :py:class:`~.Field` such as :ref:`QQ`, division (with nonzero
+        divisor) is always exact so in that case ``/`` can be used instead of
+        :py:meth:`~.Domain.exquo`.
+
+        >>> from sympy import QQ
+        >>> QQ.exquo(QQ(5), QQ(2))
+        5/2
+        >>> QQ(5) / QQ(2)
+        5/2
+
+        Parameters
+        ==========
+
+        a: domain element
+            The dividend
+        b: domain element
+            The divisor
+
+        Returns
+        =======
+
+        q: domain element
+            The exact quotient
+
+        Raises
+        ======
+
+        ExactQuotientFailed: if exact division is not possible.
+        ZeroDivisionError: when the divisor is zero.
+
+        See also
+        ========
+
+        quo: Analogue of ``a // b``
+        rem: Analogue of ``a % b``
+        div: Analogue of ``divmod(a, b)``
+
+        Notes
+        =====
+
+        Since the default :py:attr:`~.Domain.dtype` for :ref:`ZZ` is ``int``
+        (or ``mpz``) division as ``a / b`` should not be used as it would give
+        a ``float`` which is not a domain element.
+
+        >>> ZZ(4) / ZZ(2) # doctest: +SKIP
+        2.0
+        >>> ZZ(5) / ZZ(2) # doctest: +SKIP
+        2.5
+
+        On the other hand with `SYMPY_GROUND_TYPES=flint` elements of :ref:`ZZ`
+        are ``flint.fmpz`` and division would raise an exception:
+
+        >>> ZZ(4) / ZZ(2) # doctest: +SKIP
+        Traceback (most recent call last):
+        ...
+        TypeError: unsupported operand type(s) for /: 'fmpz' and 'fmpz'
+
+        Using ``/`` with :ref:`ZZ` will lead to incorrect results so
+        :py:meth:`~.Domain.exquo` should be used instead.
+
+        """
+        raise NotImplementedError
+
+    def quo(self, a, b):
+        """Quotient of *a* and *b*. Analogue of ``a // b``.
+
+        ``K.quo(a, b)`` is equivalent to ``K.div(a, b)[0]``. See
+        :py:meth:`~.Domain.div` for more explanation.
+
+        See also
+        ========
+
+        rem: Analogue of ``a % b``
+        div: Analogue of ``divmod(a, b)``
+        exquo: Analogue of ``a / b``
+        """
+        raise NotImplementedError
+
+    def rem(self, a, b):
+        """Modulo division of *a* and *b*. Analogue of ``a % b``.
+
+        ``K.rem(a, b)`` is equivalent to ``K.div(a, b)[1]``. See
+        :py:meth:`~.Domain.div` for more explanation.
+
+        See also
+        ========
+
+        quo: Analogue of ``a // b``
+        div: Analogue of ``divmod(a, b)``
+        exquo: Analogue of ``a / b``
+        """
+        raise NotImplementedError
+
+    def div(self, a, b):
+        """Quotient and remainder for *a* and *b*. Analogue of ``divmod(a, b)``
+
+        Explanation
+        ===========
+
+        This is essentially the same as ``divmod(a, b)`` except that is more
+        consistent when working over some :py:class:`~.Field` domains such as
+        :ref:`QQ`. When working over an arbitrary :py:class:`~.Domain` the
+        :py:meth:`~.Domain.div` method should be used instead of ``divmod``.
+
+        The key invariant is that if ``q, r = K.div(a, b)`` then
+        ``a == b*q + r``.
+
+        The result of ``K.div(a, b)`` is the same as the tuple
+        ``(K.quo(a, b), K.rem(a, b))`` except that if both quotient and
+        remainder are needed then it is more efficient to use
+        :py:meth:`~.Domain.div`.
+
+        Examples
+        ========
+
+        We can use ``K.div`` instead of ``divmod`` for floor division and
+        remainder.
+
+        >>> from sympy import ZZ, QQ
+        >>> ZZ.div(ZZ(5), ZZ(2))
+        (2, 1)
+
+        If ``K`` is a :py:class:`~.Field` then the division is always exact
+        with a remainder of :py:attr:`~.Domain.zero`.
+
+        >>> QQ.div(QQ(5), QQ(2))
+        (5/2, 0)
+
+        Parameters
+        ==========
+
+        a: domain element
+            The dividend
+        b: domain element
+            The divisor
+
+        Returns
+        =======
+
+        (q, r): tuple of domain elements
+            The quotient and remainder
+
+        Raises
+        ======
+
+        ZeroDivisionError: when the divisor is zero.
+
+        See also
+        ========
+
+        quo: Analogue of ``a // b``
+        rem: Analogue of ``a % b``
+        exquo: Analogue of ``a / b``
+
+        Notes
+        =====
+
+        If ``gmpy`` is installed then the ``gmpy.mpq`` type will be used as
+        the :py:attr:`~.Domain.dtype` for :ref:`QQ`. The ``gmpy.mpq`` type
+        defines ``divmod`` in a way that is undesirable so
+        :py:meth:`~.Domain.div` should be used instead of ``divmod``.
+
+        >>> a = QQ(1)
+        >>> b = QQ(3, 2)
+        >>> a               # doctest: +SKIP
+        mpq(1,1)
+        >>> b               # doctest: +SKIP
+        mpq(3,2)
+        >>> divmod(a, b)    # doctest: +SKIP
+        (mpz(0), mpq(1,1))
+        >>> QQ.div(a, b)    # doctest: +SKIP
+        (mpq(2,3), mpq(0,1))
+
+        Using ``//`` or ``%`` with :ref:`QQ` will lead to incorrect results so
+        :py:meth:`~.Domain.div` should be used instead.
+
+        """
+        raise NotImplementedError
+
+    def invert(self, a, b):
+        """Returns inversion of ``a mod b``, implies something. """
+        raise NotImplementedError
+
+    def revert(self, a):
+        """Returns ``a**(-1)`` if possible. """
+        raise NotImplementedError
+
+    def numer(self, a):
+        """Returns numerator of ``a``. """
+        raise NotImplementedError
+
+    def denom(self, a):
+        """Returns denominator of ``a``. """
+        raise NotImplementedError
+
+    def half_gcdex(self, a, b):
+        """Half extended GCD of ``a`` and ``b``. """
+        s, t, h = self.gcdex(a, b)
+        return s, h
+
+    def gcdex(self, a, b):
+        """Extended GCD of ``a`` and ``b``. """
+        raise NotImplementedError
+
+    def cofactors(self, a, b):
+        """Returns GCD and cofactors of ``a`` and ``b``. """
+        gcd = self.gcd(a, b)
+        cfa = self.quo(a, gcd)
+        cfb = self.quo(b, gcd)
+        return gcd, cfa, cfb
+
+    def gcd(self, a, b):
+        """Returns GCD of ``a`` and ``b``. """
+        raise NotImplementedError
+
+    def lcm(self, a, b):
+        """Returns LCM of ``a`` and ``b``. """
+        raise NotImplementedError
+
+    def log(self, a, b):
+        """Returns b-base logarithm of ``a``. """
+        raise NotImplementedError
+
+    def sqrt(self, a):
+        """Returns a (possibly inexact) square root of ``a``.
+
+        Explanation
+        ===========
+        There is no universal definition of "inexact square root" for all
+        domains. It is not recommended to implement this method for domains
+        other then :ref:`ZZ`.
+
+        See also
+        ========
+        exsqrt
+        """
+        raise NotImplementedError
+
+    def is_square(self, a):
+        """Returns whether ``a`` is a square in the domain.
+
+        Explanation
+        ===========
+        Returns ``True`` if there is an element ``b`` in the domain such that
+        ``b * b == a``, otherwise returns ``False``. For inexact domains like
+        :ref:`RR` and :ref:`CC`, a tiny difference in this equality can be
+        tolerated.
+
+        See also
+        ========
+        exsqrt
+        """
+        raise NotImplementedError
+
+    def exsqrt(self, a):
+        """Principal square root of a within the domain if ``a`` is square.
+
+        Explanation
+        ===========
+        The implementation of this method should return an element ``b`` in the
+        domain such that ``b * b == a``, or ``None`` if there is no such ``b``.
+        For inexact domains like :ref:`RR` and :ref:`CC`, a tiny difference in
+        this equality can be tolerated. The choice of a "principal" square root
+        should follow a consistent rule whenever possible.
+
+        See also
+        ========
+        sqrt, is_square
+        """
+        raise NotImplementedError
+
+    def evalf(self, a, prec=None, **options):
+        """Returns numerical approximation of ``a``. """
+        return self.to_sympy(a).evalf(prec, **options)
+
+    n = evalf
+
+    def real(self, a):
+        return a
+
+    def imag(self, a):
+        return self.zero
+
+    def almosteq(self, a, b, tolerance=None):
+        """Check if ``a`` and ``b`` are almost equal. """
+        return a == b
+
+    def characteristic(self):
+        """Return the characteristic of this domain. """
+        raise NotImplementedError('characteristic()')
+
+
+__all__ = ['Domain']

.venv/lib/python3.13/site-packages/sympy/polys/domains/domainelement.py

@@ -0,0 +1,38 @@
+"""Trait for implementing domain elements. """
+
+
+from sympy.utilities import public
+
+@public
+class DomainElement:
+    """
+    Represents an element of a domain.
+
+    Mix in this trait into a class whose instances should be recognized as
+    elements of a domain. Method ``parent()`` gives that domain.
+    """
+
+    __slots__ = ()
+
+    def parent(self):
+        """Get the domain associated with ``self``
+
+        Examples
+        ========
+
+        >>> from sympy import ZZ, symbols
+        >>> x, y = symbols('x, y')
+        >>> K = ZZ[x,y]
+        >>> p = K(x)**2 + K(y)**2
+        >>> p
+        x**2 + y**2
+        >>> p.parent()
+        ZZ[x,y]
+
+        Notes
+        =====
+
+        This is used by :py:meth:`~.Domain.convert` to identify the domain
+        associated with a domain element.
+        """
+        raise NotImplementedError("abstract method")

.venv/lib/python3.13/site-packages/sympy/polys/domains/expressiondomain.py

@@ -0,0 +1,278 @@
+"""Implementation of :class:`ExpressionDomain` class. """
+
+
+from sympy.core import sympify, SympifyError
+from sympy.polys.domains.domainelement import DomainElement
+from sympy.polys.domains.characteristiczero import CharacteristicZero
+from sympy.polys.domains.field import Field
+from sympy.polys.domains.simpledomain import SimpleDomain
+from sympy.polys.polyutils import PicklableWithSlots
+from sympy.utilities import public
+
+eflags = {"deep": False, "mul": True, "power_exp": False, "power_base": False,
+              "basic": False, "multinomial": False, "log": False}
+
+@public
+class ExpressionDomain(Field, CharacteristicZero, SimpleDomain):
+    """A class for arbitrary expressions. """
+
+    is_SymbolicDomain = is_EX = True
+
+    class Expression(DomainElement, PicklableWithSlots):
+        """An arbitrary expression. """
+
+        __slots__ = ('ex',)
+
+        def __init__(self, ex):
+            if not isinstance(ex, self.__class__):
+                self.ex = sympify(ex)
+            else:
+                self.ex = ex.ex
+
+        def __repr__(f):
+            return 'EX(%s)' % repr(f.ex)
+
+        def __str__(f):
+            return 'EX(%s)' % str(f.ex)
+
+        def __hash__(self):
+            return hash((self.__class__.__name__, self.ex))
+
+        def parent(self):
+            return EX
+
+        def as_expr(f):
+            return f.ex
+
+        def numer(f):
+            return f.__class__(f.ex.as_numer_denom()[0])
+
+        def denom(f):
+            return f.__class__(f.ex.as_numer_denom()[1])
+
+        def simplify(f, ex):
+            return f.__class__(ex.cancel().expand(**eflags))
+
+        def __abs__(f):
+            return f.__class__(abs(f.ex))
+
+        def __neg__(f):
+            return f.__class__(-f.ex)
+
+        def _to_ex(f, g):
+            try:
+                return f.__class__(g)
+            except SympifyError:
+                return None
+
+        def __lt__(f, g):
+            return f.ex.sort_key() < g.ex.sort_key()
+
+        def __add__(f, g):
+            g = f._to_ex(g)
+
+            if g is None:
+                return NotImplemented
+            elif g == EX.zero:
+                return f
+            elif f == EX.zero:
+                return g
+            else:
+                return f.simplify(f.ex + g.ex)
+
+        def __radd__(f, g):
+            return f.simplify(f.__class__(g).ex + f.ex)
+
+        def __sub__(f, g):
+            g = f._to_ex(g)
+
+            if g is None:
+                return NotImplemented
+            elif g == EX.zero:
+                return f
+            elif f == EX.zero:
+                return -g
+            else:
+                return f.simplify(f.ex - g.ex)
+
+        def __rsub__(f, g):
+            return f.simplify(f.__class__(g).ex - f.ex)
+
+        def __mul__(f, g):
+            g = f._to_ex(g)
+
+            if g is None:
+                return NotImplemented
+
+            if EX.zero in (f, g):
+                return EX.zero
+            elif f.ex.is_Number and g.ex.is_Number:
+                return f.__class__(f.ex*g.ex)
+
+            return f.simplify(f.ex*g.ex)
+
+        def __rmul__(f, g):
+            return f.simplify(f.__class__(g).ex*f.ex)
+
+        def __pow__(f, n):
+            n = f._to_ex(n)
+
+            if n is not None:
+                return f.simplify(f.ex**n.ex)
+            else:
+                return NotImplemented
+
+        def __truediv__(f, g):
+            g = f._to_ex(g)
+
+            if g is not None:
+                return f.simplify(f.ex/g.ex)
+            else:
+                return NotImplemented
+
+        def __rtruediv__(f, g):
+            return f.simplify(f.__class__(g).ex/f.ex)
+
+        def __eq__(f, g):
+            return f.ex == f.__class__(g).ex
+
+        def __ne__(f, g):
+            return not f == g
+
+        def __bool__(f):
+            return not f.ex.is_zero
+
+        def gcd(f, g):
+            from sympy.polys import gcd
+            return f.__class__(gcd(f.ex, f.__class__(g).ex))
+
+        def lcm(f, g):
+            from sympy.polys import lcm
+            return f.__class__(lcm(f.ex, f.__class__(g).ex))
+
+    dtype = Expression
+
+    zero = Expression(0)
+    one = Expression(1)
+
+    rep = 'EX'
+
+    has_assoc_Ring = False
+    has_assoc_Field = True
+
+    def __init__(self):
+        pass
+
+    def __eq__(self, other):
+        if isinstance(other, ExpressionDomain):
+            return True
+        else:
+            return NotImplemented
+
+    def __hash__(self):
+        return hash("EX")
+
+    def to_sympy(self, a):
+        """Convert ``a`` to a SymPy object. """
+        return a.as_expr()
+
+    def from_sympy(self, a):
+        """Convert SymPy's expression to ``dtype``. """
+        return self.dtype(a)
+
+    def from_ZZ(K1, a, K0):
+        """Convert a Python ``int`` object to ``dtype``. """
+        return K1(K0.to_sympy(a))
+
+    def from_ZZ_python(K1, a, K0):
+        """Convert a Python ``int`` object to ``dtype``. """
+        return K1(K0.to_sympy(a))
+
+    def from_QQ(K1, a, K0):
+        """Convert a Python ``Fraction`` object to ``dtype``. """
+        return K1(K0.to_sympy(a))
+
+    def from_QQ_python(K1, a, K0):
+        """Convert a Python ``Fraction`` object to ``dtype``. """
+        return K1(K0.to_sympy(a))
+
+    def from_ZZ_gmpy(K1, a, K0):
+        """Convert a GMPY ``mpz`` object to ``dtype``. """
+        return K1(K0.to_sympy(a))
+
+    def from_QQ_gmpy(K1, a, K0):
+        """Convert a GMPY ``mpq`` object to ``dtype``. """
+        return K1(K0.to_sympy(a))
+
+    def from_GaussianIntegerRing(K1, a, K0):
+        """Convert a ``GaussianRational`` object to ``dtype``. """
+        return K1(K0.to_sympy(a))
+
+    def from_GaussianRationalField(K1, a, K0):
+        """Convert a ``GaussianRational`` object to ``dtype``. """
+        return K1(K0.to_sympy(a))
+
+    def from_AlgebraicField(K1, a, K0):
+        """Convert an ``ANP`` object to ``dtype``. """
+        return K1(K0.to_sympy(a))
+
+    def from_RealField(K1, a, K0):
+        """Convert a mpmath ``mpf`` object to ``dtype``. """
+        return K1(K0.to_sympy(a))
+
+    def from_ComplexField(K1, a, K0):
+        """Convert a mpmath ``mpc`` object to ``dtype``. """
+        return K1(K0.to_sympy(a))
+
+    def from_PolynomialRing(K1, a, K0):
+        """Convert a ``DMP`` object to ``dtype``. """
+        return K1(K0.to_sympy(a))
+
+    def from_FractionField(K1, a, K0):
+        """Convert a ``DMF`` object to ``dtype``. """
+        return K1(K0.to_sympy(a))
+
+    def from_ExpressionDomain(K1, a, K0):
+        """Convert a ``EX`` object to ``dtype``. """
+        return a
+
+    def get_ring(self):
+        """Returns a ring associated with ``self``. """
+        return self  # XXX: EX is not a ring but we don't have much choice here.
+
+    def get_field(self):
+        """Returns a field associated with ``self``. """
+        return self
+
+    def is_positive(self, a):
+        """Returns True if ``a`` is positive. """
+        return a.ex.as_coeff_mul()[0].is_positive
+
+    def is_negative(self, a):
+        """Returns True if ``a`` is negative. """
+        return a.ex.could_extract_minus_sign()
+
+    def is_nonpositive(self, a):
+        """Returns True if ``a`` is non-positive. """
+        return a.ex.as_coeff_mul()[0].is_nonpositive
+
+    def is_nonnegative(self, a):
+        """Returns True if ``a`` is non-negative. """
+        return a.ex.as_coeff_mul()[0].is_nonnegative
+
+    def numer(self, a):
+        """Returns numerator of ``a``. """
+        return a.numer()
+
+    def denom(self, a):
+        """Returns denominator of ``a``. """
+        return a.denom()
+
+    def gcd(self, a, b):
+        return self(1)
+
+    def lcm(self, a, b):
+        return a.lcm(b)
+
+
+EX = ExpressionDomain()

.venv/lib/python3.13/site-packages/sympy/polys/domains/expressionrawdomain.py

@@ -0,0 +1,57 @@
+"""Implementation of :class:`ExpressionRawDomain` class. """
+
+
+from sympy.core import Expr, S, sympify, Add
+from sympy.polys.domains.characteristiczero import CharacteristicZero
+from sympy.polys.domains.field import Field
+from sympy.polys.domains.simpledomain import SimpleDomain
+from sympy.polys.polyerrors import CoercionFailed
+from sympy.utilities import public
+
+
+@public
+class ExpressionRawDomain(Field, CharacteristicZero, SimpleDomain):
+    """A class for arbitrary expressions but without automatic simplification. """
+
+    is_SymbolicRawDomain = is_EXRAW = True
+
+    dtype = Expr
+
+    zero = S.Zero
+    one = S.One
+
+    rep = 'EXRAW'
+
+    has_assoc_Ring = False
+    has_assoc_Field = True
+
+    def __init__(self):
+        pass
+
+    @classmethod
+    def new(self, a):
+        return sympify(a)
+
+    def to_sympy(self, a):
+        """Convert ``a`` to a SymPy object. """
+        return a
+
+    def from_sympy(self, a):
+        """Convert SymPy's expression to ``dtype``. """
+        if not isinstance(a, Expr):
+            raise CoercionFailed(f"Expecting an Expr instance but found: {type(a).__name__}")
+        return a
+
+    def convert_from(self, a, K):
+        """Convert a domain element from another domain to EXRAW"""
+        return K.to_sympy(a)
+
+    def get_field(self):
+        """Returns a field associated with ``self``. """
+        return self
+
+    def sum(self, items):
+        return Add(*items)
+
+
+EXRAW = ExpressionRawDomain()

.venv/lib/python3.13/site-packages/sympy/polys/domains/field.py

@@ -0,0 +1,118 @@
+"""Implementation of :class:`Field` class. """
+
+
+from sympy.polys.domains.ring import Ring
+from sympy.polys.polyerrors import NotReversible, DomainError
+from sympy.utilities import public
+
+@public
+class Field(Ring):
+    """Represents a field domain. """
+
+    is_Field = True
+    is_PID = True
+
+    def get_ring(self):
+        """Returns a ring associated with ``self``. """
+        raise DomainError('there is no ring associated with %s' % self)
+
+    def get_field(self):
+        """Returns a field associated with ``self``. """
+        return self
+
+    def exquo(self, a, b):
+        """Exact quotient of ``a`` and ``b``, implies ``__truediv__``.  """
+        return a / b
+
+    def quo(self, a, b):
+        """Quotient of ``a`` and ``b``, implies ``__truediv__``. """
+        return a / b
+
+    def rem(self, a, b):
+        """Remainder of ``a`` and ``b``, implies nothing.  """
+        return self.zero
+
+    def div(self, a, b):
+        """Division of ``a`` and ``b``, implies ``__truediv__``. """
+        return a / b, self.zero
+
+    def gcd(self, a, b):
+        """
+        Returns GCD of ``a`` and ``b``.
+
+        This definition of GCD over fields allows to clear denominators
+        in `primitive()`.
+
+        Examples
+        ========
+
+        >>> from sympy.polys.domains import QQ
+        >>> from sympy import S, gcd, primitive
+        >>> from sympy.abc import x
+
+        >>> QQ.gcd(QQ(2, 3), QQ(4, 9))
+        2/9
+        >>> gcd(S(2)/3, S(4)/9)
+        2/9
+        >>> primitive(2*x/3 + S(4)/9)
+        (2/9, 3*x + 2)
+
+        """
+        try:
+            ring = self.get_ring()
+        except DomainError:
+            return self.one
+
+        p = ring.gcd(self.numer(a), self.numer(b))
+        q = ring.lcm(self.denom(a), self.denom(b))
+
+        return self.convert(p, ring)/q
+
+    def gcdex(self, a, b):
+        """
+        Returns x, y, g such that a * x + b * y == g == gcd(a, b)
+        """
+        d = self.gcd(a, b)
+
+        if a == self.zero:
+            if b == self.zero:
+                return self.zero, self.one, self.zero
+            else:
+                return self.zero, d/b, d
+        else:
+            return d/a, self.zero, d
+
+    def lcm(self, a, b):
+        """
+        Returns LCM of ``a`` and ``b``.
+
+        >>> from sympy.polys.domains import QQ
+        >>> from sympy import S, lcm
+
+        >>> QQ.lcm(QQ(2, 3), QQ(4, 9))
+        4/3
+        >>> lcm(S(2)/3, S(4)/9)
+        4/3
+
+        """
+
+        try:
+            ring = self.get_ring()
+        except DomainError:
+            return a*b
+
+        p = ring.lcm(self.numer(a), self.numer(b))
+        q = ring.gcd(self.denom(a), self.denom(b))
+
+        return self.convert(p, ring)/q
+
+    def revert(self, a):
+        """Returns ``a**(-1)`` if possible. """
+        if a:
+            return 1/a
+        else:
+            raise NotReversible('zero is not reversible')
+
+    def is_unit(self, a):
+        """Return true if ``a`` is a invertible"""
+        return bool(a)

.venv/lib/python3.13/site-packages/sympy/polys/domains/finitefield.py

@@ -0,0 +1,368 @@
+"""Implementation of :class:`FiniteField` class. """
+
+import operator
+
+from sympy.external.gmpy import GROUND_TYPES
+from sympy.utilities.decorator import doctest_depends_on
+
+from sympy.core.numbers import int_valued
+from sympy.polys.domains.field import Field
+
+from sympy.polys.domains.modularinteger import ModularIntegerFactory
+from sympy.polys.domains.simpledomain import SimpleDomain
+from sympy.polys.galoistools import gf_zassenhaus, gf_irred_p_rabin
+from sympy.polys.polyerrors import CoercionFailed
+from sympy.utilities import public
+from sympy.polys.domains.groundtypes import SymPyInteger
+
+
+if GROUND_TYPES == 'flint':
+    __doctest_skip__ = ['FiniteField']
+
+
+if GROUND_TYPES == 'flint':
+    import flint
+    # Don't use python-flint < 0.5.0 because nmod was missing some features in
+    # previous versions of python-flint and fmpz_mod was not yet added.
+    _major, _minor, *_ = flint.__version__.split('.')
+    if (int(_major), int(_minor)) < (0, 5):
+        flint = None
+else:
+    flint = None
+
+
+def _modular_int_factory_nmod(mod):
+    # nmod only recognises int
+    index = operator.index
+    mod = index(mod)
+    nmod = flint.nmod
+    nmod_poly = flint.nmod_poly
+
+    # flint's nmod is only for moduli up to 2^64-1 (on a 64-bit machine)
+    try:
+        nmod(0, mod)
+    except OverflowError:
+        return None, None
+
+    def ctx(x):
+        try:
+            return nmod(x, mod)
+        except TypeError:
+            return nmod(index(x), mod)
+
+    def poly_ctx(cs):
+        return nmod_poly(cs, mod)
+
+    return ctx, poly_ctx
+
+
+def _modular_int_factory_fmpz_mod(mod):
+    index = operator.index
+    fctx = flint.fmpz_mod_ctx(mod)
+    fctx_poly = flint.fmpz_mod_poly_ctx(mod)
+    fmpz_mod_poly = flint.fmpz_mod_poly
+
+    def ctx(x):
+        try:
+            return fctx(x)
+        except TypeError:
+            # x might be Integer
+            return fctx(index(x))
+
+    def poly_ctx(cs):
+        return fmpz_mod_poly(cs, fctx_poly)
+
+    return ctx, poly_ctx
+
+
+def _modular_int_factory(mod, dom, symmetric, self):
+    # Convert the modulus to ZZ
+    try:
+        mod = dom.convert(mod)
+    except CoercionFailed:
+        raise ValueError('modulus must be an integer, got %s' % mod)
+
+    ctx, poly_ctx, is_flint = None, None, False
+
+    # Don't use flint if the modulus is not prime as it often crashes.
+    if flint is not None and mod.is_prime():
+
+        is_flint = True
+
+        # Try to use flint's nmod first
+        ctx, poly_ctx = _modular_int_factory_nmod(mod)
+
+        if ctx is None:
+            # Use fmpz_mod for larger moduli
+            ctx, poly_ctx = _modular_int_factory_fmpz_mod(mod)
+
+    if ctx is None:
+        # Use the Python implementation if flint is not available or the
+        # modulus is not prime.
+        ctx = ModularIntegerFactory(mod, dom, symmetric, self)
+        poly_ctx = None  # not used
+
+    return ctx, poly_ctx, is_flint
+
+
+@public
+@doctest_depends_on(modules=['python', 'gmpy'])
+class FiniteField(Field, SimpleDomain):
+    r"""Finite field of prime order :ref:`GF(p)`
+
+    A :ref:`GF(p)` domain represents a `finite field`_ `\mathbb{F}_p` of prime
+    order as :py:class:`~.Domain` in the domain system (see
+    :ref:`polys-domainsintro`).
+
+    A :py:class:`~.Poly` created from an expression with integer
+    coefficients will have the domain :ref:`ZZ`. However, if the ``modulus=p``
+    option is given then the domain will be a finite field instead.
+
+    >>> from sympy import Poly, Symbol
+    >>> x = Symbol('x')
+    >>> p = Poly(x**2 + 1)
+    >>> p
+    Poly(x**2 + 1, x, domain='ZZ')
+    >>> p.domain
+    ZZ
+    >>> p2 = Poly(x**2 + 1, modulus=2)
+    >>> p2
+    Poly(x**2 + 1, x, modulus=2)
+    >>> p2.domain
+    GF(2)
+
+    It is possible to factorise a polynomial over :ref:`GF(p)` using the
+    modulus argument to :py:func:`~.factor` or by specifying the domain
+    explicitly. The domain can also be given as a string.
+
+    >>> from sympy import factor, GF
+    >>> factor(x**2 + 1)
+    x**2 + 1
+    >>> factor(x**2 + 1, modulus=2)
+    (x + 1)**2
+    >>> factor(x**2 + 1, domain=GF(2))
+    (x + 1)**2
+    >>> factor(x**2 + 1, domain='GF(2)')
+    (x + 1)**2
+
+    It is also possible to use :ref:`GF(p)` with the :py:func:`~.cancel`
+    and :py:func:`~.gcd` functions.
+
+    >>> from sympy import cancel, gcd
+    >>> cancel((x**2 + 1)/(x + 1))
+    (x**2 + 1)/(x + 1)
+    >>> cancel((x**2 + 1)/(x + 1), domain=GF(2))
+    x + 1
+    >>> gcd(x**2 + 1, x + 1)
+    1
+    >>> gcd(x**2 + 1, x + 1, domain=GF(2))
+    x + 1
+
+    When using the domain directly :ref:`GF(p)` can be used as a constructor
+    to create instances which then support the operations ``+,-,*,**,/``
+
+    >>> from sympy import GF
+    >>> K = GF(5)
+    >>> K
+    GF(5)
+    >>> x = K(3)
+    >>> y = K(2)
+    >>> x
+    3 mod 5
+    >>> y
+    2 mod 5
+    >>> x * y
+    1 mod 5
+    >>> x / y
+    4 mod 5
+
+    Notes
+    =====
+
+    It is also possible to create a :ref:`GF(p)` domain of **non-prime**
+    order but the resulting ring is **not** a field: it is just the ring of
+    the integers modulo ``n``.
+
+    >>> K = GF(9)
+    >>> z = K(3)
+    >>> z
+    3 mod 9
+    >>> z**2
+    0 mod 9
+
+    It would be good to have a proper implementation of prime power fields
+    (``GF(p**n)``) but these are not yet implemented in SymPY.
+
+    .. _finite field: https://en.wikipedia.org/wiki/Finite_field
+    """
+
+    rep = 'FF'
+    alias = 'FF'
+
+    is_FiniteField = is_FF = True
+    is_Numerical = True
+
+    has_assoc_Ring = False
+    has_assoc_Field = True
+
+    dom = None
+    mod = None
+
+    def __init__(self, mod, symmetric=True):
+        from sympy.polys.domains import ZZ
+        dom = ZZ
+
+        if mod <= 0:
+            raise ValueError('modulus must be a positive integer, got %s' % mod)
+
+        ctx, poly_ctx, is_flint = _modular_int_factory(mod, dom, symmetric, self)
+
+        self.dtype = ctx
+        self._poly_ctx = poly_ctx
+        self._is_flint = is_flint
+
+        self.zero = self.dtype(0)
+        self.one = self.dtype(1)
+        self.dom = dom
+        self.mod = mod
+        self.sym = symmetric
+        self._tp = type(self.zero)
+
+    @property
+    def tp(self):
+        return self._tp
+
+    @property
+    def is_Field(self):
+        is_field = getattr(self, '_is_field', None)
+        if is_field is None:
+            from sympy.ntheory.primetest import isprime
+            self._is_field = is_field = isprime(self.mod)
+        return is_field
+
+    def __str__(self):
+        return 'GF(%s)' % self.mod
+
+    def __hash__(self):
+        return hash((self.__class__.__name__, self.dtype, self.mod, self.dom))
+
+    def __eq__(self, other):
+        """Returns ``True`` if two domains are equivalent. """
+        return isinstance(other, FiniteField) and \
+            self.mod == other.mod and self.dom == other.dom
+
+    def characteristic(self):
+        """Return the characteristic of this domain. """
+        return self.mod
+
+    def get_field(self):
+        """Returns a field associated with ``self``. """
+        return self
+
+    def to_sympy(self, a):
+        """Convert ``a`` to a SymPy object. """
+        return SymPyInteger(self.to_int(a))
+
+    def from_sympy(self, a):
+        """Convert SymPy's Integer to SymPy's ``Integer``. """
+        if a.is_Integer:
+            return self.dtype(self.dom.dtype(int(a)))
+        elif int_valued(a):
+            return self.dtype(self.dom.dtype(int(a)))
+        else:
+            raise CoercionFailed("expected an integer, got %s" % a)
+
+    def to_int(self, a):
+        """Convert ``val`` to a Python ``int`` object. """
+        aval = int(a)
+        if self.sym and aval > self.mod // 2:
+            aval -= self.mod
+        return aval
+
+    def is_positive(self, a):
+        """Returns True if ``a`` is positive. """
+        return bool(a)
+
+    def is_nonnegative(self, a):
+        """Returns True if ``a`` is non-negative. """
+        return True
+
+    def is_negative(self, a):
+        """Returns True if ``a`` is negative. """
+        return False
+
+    def is_nonpositive(self, a):
+        """Returns True if ``a`` is non-positive. """
+        return not a
+
+    def from_FF(K1, a, K0=None):
+        """Convert ``ModularInteger(int)`` to ``dtype``. """
+        return K1.dtype(K1.dom.from_ZZ(int(a), K0.dom))
+
+    def from_FF_python(K1, a, K0=None):
+        """Convert ``ModularInteger(int)`` to ``dtype``. """
+        return K1.dtype(K1.dom.from_ZZ_python(int(a), K0.dom))
+
+    def from_ZZ(K1, a, K0=None):
+        """Convert Python's ``int`` to ``dtype``. """
+        return K1.dtype(K1.dom.from_ZZ_python(a, K0))
+
+    def from_ZZ_python(K1, a, K0=None):
+        """Convert Python's ``int`` to ``dtype``. """
+        return K1.dtype(K1.dom.from_ZZ_python(a, K0))
+
+    def from_QQ(K1, a, K0=None):
+        """Convert Python's ``Fraction`` to ``dtype``. """
+        if a.denominator == 1:
+            return K1.from_ZZ_python(a.numerator)
+
+    def from_QQ_python(K1, a, K0=None):
+        """Convert Python's ``Fraction`` to ``dtype``. """
+        if a.denominator == 1:
+            return K1.from_ZZ_python(a.numerator)
+
+    def from_FF_gmpy(K1, a, K0=None):
+        """Convert ``ModularInteger(mpz)`` to ``dtype``. """
+        return K1.dtype(K1.dom.from_ZZ_gmpy(a.val, K0.dom))
+
+    def from_ZZ_gmpy(K1, a, K0=None):
+        """Convert GMPY's ``mpz`` to ``dtype``. """
+        return K1.dtype(K1.dom.from_ZZ_gmpy(a, K0))
+
+    def from_QQ_gmpy(K1, a, K0=None):
+        """Convert GMPY's ``mpq`` to ``dtype``. """
+        if a.denominator == 1:
+            return K1.from_ZZ_gmpy(a.numerator)
+
+    def from_RealField(K1, a, K0):
+        """Convert mpmath's ``mpf`` to ``dtype``. """
+        p, q = K0.to_rational(a)
+
+        if q == 1:
+            return K1.dtype(K1.dom.dtype(p))
+
+    def is_square(self, a):
+        """Returns True if ``a`` is a quadratic residue modulo p. """
+        # a is not a square <=> x**2-a is irreducible
+        poly = [int(x) for x in [self.one, self.zero, -a]]
+        return not gf_irred_p_rabin(poly, self.mod, self.dom)
+
+    def exsqrt(self, a):
+        """Square root modulo p of ``a`` if it is a quadratic residue.
+
+        Explanation
+        ===========
+        Always returns the square root that is no larger than ``p // 2``.
+        """
+        # x**2-a is not square-free if a=0 or the field is characteristic 2
+        if self.mod == 2 or a == 0:
+            return a
+        # Otherwise, use square-free factorization routine to factorize x**2-a
+        poly = [int(x) for x in [self.one, self.zero, -a]]
+        for factor in gf_zassenhaus(poly, self.mod, self.dom):
+            if len(factor) == 2 and factor[1] <= self.mod // 2:
+                return self.dtype(factor[1])
+        return None
+
+
+FF = GF = FiniteField

.venv/lib/python3.13/site-packages/sympy/polys/domains/fractionfield.py

@@ -0,0 +1,181 @@
+"""Implementation of :class:`FractionField` class. """
+
+
+from sympy.polys.domains.compositedomain import CompositeDomain
+from sympy.polys.domains.field import Field
+from sympy.polys.polyerrors import CoercionFailed, GeneratorsError
+from sympy.utilities import public
+
+@public
+class FractionField(Field, CompositeDomain):
+    """A class for representing multivariate rational function fields. """
+
+    is_FractionField = is_Frac = True
+
+    has_assoc_Ring = True
+    has_assoc_Field = True
+
+    def __init__(self, domain_or_field, symbols=None, order=None):
+        from sympy.polys.fields import FracField
+
+        if isinstance(domain_or_field, FracField) and symbols is None and order is None:
+            field = domain_or_field
+        else:
+            field = FracField(symbols, domain_or_field, order)
+
+        self.field = field
+        self.dtype = field.dtype
+
+        self.gens = field.gens
+        self.ngens = field.ngens
+        self.symbols = field.symbols
+        self.domain = field.domain
+
+        # TODO: remove this
+        self.dom = self.domain
+
+    def new(self, element):
+        return self.field.field_new(element)
+
+    def of_type(self, element):
+        """Check if ``a`` is of type ``dtype``. """
+        return self.field.is_element(element)
+
+    @property
+    def zero(self):
+        return self.field.zero
+
+    @property
+    def one(self):
+        return self.field.one
+
+    @property
+    def order(self):
+        return self.field.order
+
+    def __str__(self):
+        return str(self.domain) + '(' + ','.join(map(str, self.symbols)) + ')'
+
+    def __hash__(self):
+        return hash((self.__class__.__name__, self.field, self.domain, self.symbols))
+
+    def __eq__(self, other):
+        """Returns ``True`` if two domains are equivalent. """
+        if not isinstance(other, FractionField):
+            return NotImplemented
+        return self.field == other.field
+
+    def to_sympy(self, a):
+        """Convert ``a`` to a SymPy object. """
+        return a.as_expr()
+
+    def from_sympy(self, a):
+        """Convert SymPy's expression to ``dtype``. """
+        return self.field.from_expr(a)
+
+    def from_ZZ(K1, a, K0):
+        """Convert a Python ``int`` object to ``dtype``. """
+        return K1(K1.domain.convert(a, K0))
+
+    def from_ZZ_python(K1, a, K0):
+        """Convert a Python ``int`` object to ``dtype``. """
+        return K1(K1.domain.convert(a, K0))
+
+    def from_QQ(K1, a, K0):
+        """Convert a Python ``Fraction`` object to ``dtype``. """
+        dom = K1.domain
+        conv = dom.convert_from
+        if dom.is_ZZ:
+            return K1(conv(K0.numer(a), K0)) / K1(conv(K0.denom(a), K0))
+        else:
+            return K1(conv(a, K0))
+
+    def from_QQ_python(K1, a, K0):
+        """Convert a Python ``Fraction`` object to ``dtype``. """
+        return K1(K1.domain.convert(a, K0))
+
+    def from_ZZ_gmpy(K1, a, K0):
+        """Convert a GMPY ``mpz`` object to ``dtype``. """
+        return K1(K1.domain.convert(a, K0))
+
+    def from_QQ_gmpy(K1, a, K0):
+        """Convert a GMPY ``mpq`` object to ``dtype``. """
+        return K1(K1.domain.convert(a, K0))
+
+    def from_GaussianRationalField(K1, a, K0):
+        """Convert a ``GaussianRational`` object to ``dtype``. """
+        return K1(K1.domain.convert(a, K0))
+
+    def from_GaussianIntegerRing(K1, a, K0):
+        """Convert a ``GaussianInteger`` object to ``dtype``. """
+        return K1(K1.domain.convert(a, K0))
+
+    def from_RealField(K1, a, K0):
+        """Convert a mpmath ``mpf`` object to ``dtype``. """
+        return K1(K1.domain.convert(a, K0))
+
+    def from_ComplexField(K1, a, K0):
+        """Convert a mpmath ``mpf`` object to ``dtype``. """
+        return K1(K1.domain.convert(a, K0))
+
+    def from_AlgebraicField(K1, a, K0):
+        """Convert an algebraic number to ``dtype``. """
+        if K1.domain != K0:
+            a = K1.domain.convert_from(a, K0)
+        if a is not None:
+            return K1.new(a)
+
+    def from_PolynomialRing(K1, a, K0):
+        """Convert a polynomial to ``dtype``. """
+        if a.is_ground:
+            return K1.convert_from(a.coeff(1), K0.domain)
+        try:
+            return K1.new(a.set_ring(K1.field.ring))
+        except (CoercionFailed, GeneratorsError):
+            # XXX: We get here if K1=ZZ(x,y) and K0=QQ[x,y]
+            # and the poly a in K0 has non-integer coefficients.
+            # It seems that K1.new can handle this but K1.new doesn't work
+            # when K0.domain is an algebraic field...
+            try:
+                return K1.new(a)
+            except (CoercionFailed, GeneratorsError):
+                return None
+
+    def from_FractionField(K1, a, K0):
+        """Convert a rational function to ``dtype``. """
+        try:
+            return a.set_field(K1.field)
+        except (CoercionFailed, GeneratorsError):
+            return None
+
+    def get_ring(self):
+        """Returns a field associated with ``self``. """
+        return self.field.to_ring().to_domain()
+
+    def is_positive(self, a):
+        """Returns True if ``LC(a)`` is positive. """
+        return self.domain.is_positive(a.numer.LC)
+
+    def is_negative(self, a):
+        """Returns True if ``LC(a)`` is negative. """
+        return self.domain.is_negative(a.numer.LC)
+
+    def is_nonpositive(self, a):
+        """Returns True if ``LC(a)`` is non-positive. """
+        return self.domain.is_nonpositive(a.numer.LC)
+
+    def is_nonnegative(self, a):
+        """Returns True if ``LC(a)`` is non-negative. """
+        return self.domain.is_nonnegative(a.numer.LC)
+
+    def numer(self, a):
+        """Returns numerator of ``a``. """
+        return a.numer
+
+    def denom(self, a):
+        """Returns denominator of ``a``. """
+        return a.denom
+
+    def factorial(self, a):
+        """Returns factorial of ``a``. """
+        return self.dtype(self.domain.factorial(a))

.venv/lib/python3.13/site-packages/sympy/polys/domains/gaussiandomains.py

@@ -0,0 +1,706 @@
+"""Domains of Gaussian type."""
+
+from __future__ import annotations
+from sympy.core.numbers import I
+from sympy.polys.polyclasses import DMP
+from sympy.polys.polyerrors import CoercionFailed
+from sympy.polys.domains.integerring import ZZ
+from sympy.polys.domains.rationalfield import QQ
+from sympy.polys.domains.algebraicfield import AlgebraicField
+from sympy.polys.domains.domain import Domain
+from sympy.polys.domains.domainelement import DomainElement
+from sympy.polys.domains.field import Field
+from sympy.polys.domains.ring import Ring
+
+
+class GaussianElement(DomainElement):
+    """Base class for elements of Gaussian type domains."""
+    base: Domain
+    _parent: Domain
+
+    __slots__ = ('x', 'y')
+
+    def __new__(cls, x, y=0):
+        conv = cls.base.convert
+        return cls.new(conv(x), conv(y))
+
+    @classmethod
+    def new(cls, x, y):
+        """Create a new GaussianElement of the same domain."""
+        obj = super().__new__(cls)
+        obj.x = x
+        obj.y = y
+        return obj
+
+    def parent(self):
+        """The domain that this is an element of (ZZ_I or QQ_I)"""
+        return self._parent
+
+    def __hash__(self):
+        return hash((self.x, self.y))
+
+    def __eq__(self, other):
+        if isinstance(other, self.__class__):
+            return self.x == other.x and self.y == other.y
+        else:
+            return NotImplemented
+
+    def __lt__(self, other):
+        if not isinstance(other, GaussianElement):
+            return NotImplemented
+        return [self.y, self.x] < [other.y, other.x]
+
+    def __pos__(self):
+        return self
+
+    def __neg__(self):
+        return self.new(-self.x, -self.y)
+
+    def __repr__(self):
+        return "%s(%s, %s)" % (self._parent.rep, self.x, self.y)
+
+    def __str__(self):
+        return str(self._parent.to_sympy(self))
+
+    @classmethod
+    def _get_xy(cls, other):
+        if not isinstance(other, cls):
+            try:
+                other = cls._parent.convert(other)
+            except CoercionFailed:
+                return None, None
+        return other.x, other.y
+
+    def __add__(self, other):
+        x, y = self._get_xy(other)
+        if x is not None:
+            return self.new(self.x + x, self.y + y)
+        else:
+            return NotImplemented
+
+    __radd__ = __add__
+
+    def __sub__(self, other):
+        x, y = self._get_xy(other)
+        if x is not None:
+            return self.new(self.x - x, self.y - y)
+        else:
+            return NotImplemented
+
+    def __rsub__(self, other):
+        x, y = self._get_xy(other)
+        if x is not None:
+            return self.new(x - self.x, y - self.y)
+        else:
+            return NotImplemented
+
+    def __mul__(self, other):
+        x, y = self._get_xy(other)
+        if x is not None:
+            return self.new(self.x*x - self.y*y, self.x*y + self.y*x)
+        else:
+            return NotImplemented
+
+    __rmul__ = __mul__
+
+    def __pow__(self, exp):
+        if exp == 0:
+            return self.new(1, 0)
+        if exp < 0:
+            self, exp = 1/self, -exp
+        if exp == 1:
+            return self
+        pow2 = self
+        prod = self if exp % 2 else self._parent.one
+        exp //= 2
+        while exp:
+            pow2 *= pow2
+            if exp % 2:
+                prod *= pow2
+            exp //= 2
+        return prod
+
+    def __bool__(self):
+        return bool(self.x) or bool(self.y)
+
+    def quadrant(self):
+        """Return quadrant index 0-3.
+
+        0 is included in quadrant 0.
+        """
+        if self.y > 0:
+            return 0 if self.x > 0 else 1
+        elif self.y < 0:
+            return 2 if self.x < 0 else 3
+        else:
+            return 0 if self.x >= 0 else 2
+
+    def __rdivmod__(self, other):
+        try:
+            other = self._parent.convert(other)
+        except CoercionFailed:
+            return NotImplemented
+        else:
+            return other.__divmod__(self)
+
+    def __rtruediv__(self, other):
+        try:
+            other = QQ_I.convert(other)
+        except CoercionFailed:
+            return NotImplemented
+        else:
+            return other.__truediv__(self)
+
+    def __floordiv__(self, other):
+        qr = self.__divmod__(other)
+        return qr if qr is NotImplemented else qr[0]
+
+    def __rfloordiv__(self, other):
+        qr = self.__rdivmod__(other)
+        return qr if qr is NotImplemented else qr[0]
+
+    def __mod__(self, other):
+        qr = self.__divmod__(other)
+        return qr if qr is NotImplemented else qr[1]
+
+    def __rmod__(self, other):
+        qr = self.__rdivmod__(other)
+        return qr if qr is NotImplemented else qr[1]
+
+
+class GaussianInteger(GaussianElement):
+    """Gaussian integer: domain element for :ref:`ZZ_I`
+
+        >>> from sympy import ZZ_I
+        >>> z = ZZ_I(2, 3)
+        >>> z
+        (2 + 3*I)
+        >>> type(z)
+        <class 'sympy.polys.domains.gaussiandomains.GaussianInteger'>
+    """
+    base = ZZ
+
+    def __truediv__(self, other):
+        """Return a Gaussian rational."""
+        return QQ_I.convert(self)/other
+
+    def __divmod__(self, other):
+        if not other:
+            raise ZeroDivisionError('divmod({}, 0)'.format(self))
+        x, y = self._get_xy(other)
+        if x is None:
+            return NotImplemented
+
+        # multiply self and other by x - I*y
+        # self/other == (a + I*b)/c
+        a, b = self.x*x + self.y*y, -self.x*y + self.y*x
+        c = x*x + y*y
+
+        # find integers qx and qy such that
+        # |a - qx*c| <= c/2 and |b - qy*c| <= c/2
+        qx = (2*a + c) // (2*c)  # -c <= 2*a - qx*2*c < c
+        qy = (2*b + c) // (2*c)
+
+        q = GaussianInteger(qx, qy)
+        # |self/other - q| < 1 since
+        # |a/c - qx|**2 + |b/c - qy|**2 <= 1/4 + 1/4 < 1
+
+        return q, self - q*other  # |r| < |other|
+
+
+class GaussianRational(GaussianElement):
+    """Gaussian rational: domain element for :ref:`QQ_I`
+
+        >>> from sympy import QQ_I, QQ
+        >>> z = QQ_I(QQ(2, 3), QQ(4, 5))
+        >>> z
+        (2/3 + 4/5*I)
+        >>> type(z)
+        <class 'sympy.polys.domains.gaussiandomains.GaussianRational'>
+    """
+    base = QQ
+
+    def __truediv__(self, other):
+        """Return a Gaussian rational."""
+        if not other:
+            raise ZeroDivisionError('{} / 0'.format(self))
+        x, y = self._get_xy(other)
+        if x is None:
+            return NotImplemented
+        c = x*x + y*y
+
+        return GaussianRational((self.x*x + self.y*y)/c,
+                                (-self.x*y + self.y*x)/c)
+
+    def __divmod__(self, other):
+        try:
+            other = self._parent.convert(other)
+        except CoercionFailed:
+            return NotImplemented
+        if not other:
+            raise ZeroDivisionError('{} % 0'.format(self))
+        else:
+            return self/other, QQ_I.zero
+
+
+class GaussianDomain():
+    """Base class for Gaussian domains."""
+    dom: Domain
+
+    is_Numerical = True
+    is_Exact = True
+
+    has_assoc_Ring = True
+    has_assoc_Field = True
+
+    def to_sympy(self, a):
+        """Convert ``a`` to a SymPy object. """
+        conv = self.dom.to_sympy
+        return conv(a.x) + I*conv(a.y)
+
+    def from_sympy(self, a):
+        """Convert a SymPy object to ``self.dtype``."""
+        r, b = a.as_coeff_Add()
+        x = self.dom.from_sympy(r)  # may raise CoercionFailed
+        if not b:
+            return self.new(x, 0)
+        r, b = b.as_coeff_Mul()
+        y = self.dom.from_sympy(r)
+        if b is I:
+            return self.new(x, y)
+        else:
+            raise CoercionFailed("{} is not Gaussian".format(a))
+
+    def inject(self, *gens):
+        """Inject generators into this domain. """
+        return self.poly_ring(*gens)
+
+    def canonical_unit(self, d):
+        unit = self.units[-d.quadrant()]  # - for inverse power
+        return unit
+
+    def is_negative(self, element):
+        """Returns ``False`` for any ``GaussianElement``. """
+        return False
+
+    def is_positive(self, element):
+        """Returns ``False`` for any ``GaussianElement``. """
+        return False
+
+    def is_nonnegative(self, element):
+        """Returns ``False`` for any ``GaussianElement``. """
+        return False
+
+    def is_nonpositive(self, element):
+        """Returns ``False`` for any ``GaussianElement``. """
+        return False
+
+    def from_ZZ_gmpy(K1, a, K0):
+        """Convert a GMPY mpz to ``self.dtype``."""
+        return K1(a)
+
+    def from_ZZ(K1, a, K0):
+        """Convert a ZZ_python element to ``self.dtype``."""
+        return K1(a)
+
+    def from_ZZ_python(K1, a, K0):
+        """Convert a ZZ_python element to ``self.dtype``."""
+        return K1(a)
+
+    def from_QQ(K1, a, K0):
+        """Convert a GMPY mpq to ``self.dtype``."""
+        return K1(a)
+
+    def from_QQ_gmpy(K1, a, K0):
+        """Convert a GMPY mpq to ``self.dtype``."""
+        return K1(a)
+
+    def from_QQ_python(K1, a, K0):
+        """Convert a QQ_python element to ``self.dtype``."""
+        return K1(a)
+
+    def from_AlgebraicField(K1, a, K0):
+        """Convert an element from ZZ<I> or QQ<I> to ``self.dtype``."""
+        if K0.ext.args[0] == I:
+            return K1.from_sympy(K0.to_sympy(a))
+
+
+class GaussianIntegerRing(GaussianDomain, Ring):
+    r"""Ring of Gaussian integers ``ZZ_I``
+
+    The :ref:`ZZ_I` domain represents the `Gaussian integers`_ `\mathbb{Z}[i]`
+    as a :py:class:`~.Domain` in the domain system (see
+    :ref:`polys-domainsintro`).
+
+    By default a :py:class:`~.Poly` created from an expression with
+    coefficients that are combinations of integers and ``I`` (`\sqrt{-1}`)
+    will have the domain :ref:`ZZ_I`.
+
+    >>> from sympy import Poly, Symbol, I
+    >>> x = Symbol('x')
+    >>> p = Poly(x**2 + I)
+    >>> p
+    Poly(x**2 + I, x, domain='ZZ_I')
+    >>> p.domain
+    ZZ_I
+
+    The :ref:`ZZ_I` domain can be used to factorise polynomials that are
+    reducible over the Gaussian integers.
+
+    >>> from sympy import factor
+    >>> factor(x**2 + 1)
+    x**2 + 1
+    >>> factor(x**2 + 1, domain='ZZ_I')
+    (x - I)*(x + I)
+
+    The corresponding `field of fractions`_ is the domain of the Gaussian
+    rationals :ref:`QQ_I`. Conversely :ref:`ZZ_I` is the `ring of integers`_
+    of :ref:`QQ_I`.
+
+    >>> from sympy import ZZ_I, QQ_I
+    >>> ZZ_I.get_field()
+    QQ_I
+    >>> QQ_I.get_ring()
+    ZZ_I
+
+    When using the domain directly :ref:`ZZ_I` can be used as a constructor.
+
+    >>> ZZ_I(3, 4)
+    (3 + 4*I)
+    >>> ZZ_I(5)
+    (5 + 0*I)
+
+    The domain elements of :ref:`ZZ_I` are instances of
+    :py:class:`~.GaussianInteger` which support the rings operations
+    ``+,-,*,**``.
+
+    >>> z1 = ZZ_I(5, 1)
+    >>> z2 = ZZ_I(2, 3)
+    >>> z1
+    (5 + 1*I)
+    >>> z2
+    (2 + 3*I)
+    >>> z1 + z2
+    (7 + 4*I)
+    >>> z1 * z2
+    (7 + 17*I)
+    >>> z1 ** 2
+    (24 + 10*I)
+
+    Both floor (``//``) and modulo (``%``) division work with
+    :py:class:`~.GaussianInteger` (see the :py:meth:`~.Domain.div` method).
+
+    >>> z3, z4 = ZZ_I(5), ZZ_I(1, 3)
+    >>> z3 // z4  # floor division
+    (1 + -1*I)
+    >>> z3 % z4   # modulo division (remainder)
+    (1 + -2*I)
+    >>> (z3//z4)*z4 + z3%z4 == z3
+    True
+
+    True division (``/``) in :ref:`ZZ_I` gives an element of :ref:`QQ_I`. The
+    :py:meth:`~.Domain.exquo` method can be used to divide in :ref:`ZZ_I` when
+    exact division is possible.
+
+    >>> z1 / z2
+    (1 + -1*I)
+    >>> ZZ_I.exquo(z1, z2)
+    (1 + -1*I)
+    >>> z3 / z4
+    (1/2 + -3/2*I)
+    >>> ZZ_I.exquo(z3, z4)
+    Traceback (most recent call last):
+        ...
+    ExactQuotientFailed: (1 + 3*I) does not divide (5 + 0*I) in ZZ_I
+
+    The :py:meth:`~.Domain.gcd` method can be used to compute the `gcd`_ of any
+    two elements.
+
+    >>> ZZ_I.gcd(ZZ_I(10), ZZ_I(2))
+    (2 + 0*I)
+    >>> ZZ_I.gcd(ZZ_I(5), ZZ_I(2, 1))
+    (2 + 1*I)
+
+    .. _Gaussian integers: https://en.wikipedia.org/wiki/Gaussian_integer
+    .. _gcd: https://en.wikipedia.org/wiki/Greatest_common_divisor
+
+    """
+    dom = ZZ
+    mod = DMP([ZZ.one, ZZ.zero, ZZ.one], ZZ)
+    dtype = GaussianInteger
+    zero = dtype(ZZ(0), ZZ(0))
+    one = dtype(ZZ(1), ZZ(0))
+    imag_unit = dtype(ZZ(0), ZZ(1))
+    units = (one, imag_unit, -one, -imag_unit)  # powers of i
+
+    rep = 'ZZ_I'
+
+    is_GaussianRing = True
+    is_ZZ_I = True
+    is_PID = True
+
+    def __init__(self):  # override Domain.__init__
+        """For constructing ZZ_I."""
+
+    def __eq__(self, other):
+        """Returns ``True`` if two domains are equivalent. """
+        if isinstance(other, GaussianIntegerRing):
+            return True
+        else:
+            return NotImplemented
+
+    def __hash__(self):
+        """Compute hash code of ``self``. """
+        return hash('ZZ_I')
+
+    @property
+    def has_CharacteristicZero(self):
+        return True
+
+    def characteristic(self):
+        return 0
+
+    def get_ring(self):
+        """Returns a ring associated with ``self``. """
+        return self
+
+    def get_field(self):
+        """Returns a field associated with ``self``. """
+        return QQ_I
+
+    def normalize(self, d, *args):
+        """Return first quadrant element associated with ``d``.
+
+        Also multiply the other arguments by the same power of i.
+        """
+        unit = self.canonical_unit(d)
+        d *= unit
+        args = tuple(a*unit for a in args)
+        return (d,) + args if args else d
+
+    def gcd(self, a, b):
+        """Greatest common divisor of a and b over ZZ_I."""
+        while b:
+            a, b = b, a % b
+        return self.normalize(a)
+
+    def gcdex(self, a, b):
+        """Return x, y, g such that x * a + y * b = g = gcd(a, b)"""
+        x_a = self.one
+        x_b = self.zero
+        y_a = self.zero
+        y_b = self.one
+        while b:
+            q = a // b
+            a, b = b, a - q * b
+            x_a, x_b = x_b, x_a - q * x_b
+            y_a, y_b = y_b, y_a - q * y_b
+
+        a, x_a, y_a = self.normalize(a, x_a, y_a)
+        return x_a, y_a, a
+
+    def lcm(self, a, b):
+        """Least common multiple of a and b over ZZ_I."""
+        return (a * b) // self.gcd(a, b)
+
+    def from_GaussianIntegerRing(K1, a, K0):
+        """Convert a ZZ_I element to ZZ_I."""
+        return a
+
+    def from_GaussianRationalField(K1, a, K0):
+        """Convert a QQ_I element to ZZ_I."""
+        return K1.new(ZZ.convert(a.x), ZZ.convert(a.y))
+
+ZZ_I = GaussianInteger._parent = GaussianIntegerRing()
+
+
+class GaussianRationalField(GaussianDomain, Field):
+    r"""Field of Gaussian rationals ``QQ_I``
+
+    The :ref:`QQ_I` domain represents the `Gaussian rationals`_ `\mathbb{Q}(i)`
+    as a :py:class:`~.Domain` in the domain system (see
+    :ref:`polys-domainsintro`).
+
+    By default a :py:class:`~.Poly` created from an expression with
+    coefficients that are combinations of rationals and ``I`` (`\sqrt{-1}`)
+    will have the domain :ref:`QQ_I`.
+
+    >>> from sympy import Poly, Symbol, I
+    >>> x = Symbol('x')
+    >>> p = Poly(x**2 + I/2)
+    >>> p
+    Poly(x**2 + I/2, x, domain='QQ_I')
+    >>> p.domain
+    QQ_I
+
+    The polys option ``gaussian=True`` can be used to specify that the domain
+    should be :ref:`QQ_I` even if the coefficients do not contain ``I`` or are
+    all integers.
+
+    >>> Poly(x**2)
+    Poly(x**2, x, domain='ZZ')
+    >>> Poly(x**2 + I)
+    Poly(x**2 + I, x, domain='ZZ_I')
+    >>> Poly(x**2/2)
+    Poly(1/2*x**2, x, domain='QQ')
+    >>> Poly(x**2, gaussian=True)
+    Poly(x**2, x, domain='QQ_I')
+    >>> Poly(x**2 + I, gaussian=True)
+    Poly(x**2 + I, x, domain='QQ_I')
+    >>> Poly(x**2/2, gaussian=True)
+    Poly(1/2*x**2, x, domain='QQ_I')
+
+    The :ref:`QQ_I` domain can be used to factorise polynomials that are
+    reducible over the Gaussian rationals.
+
+    >>> from sympy import factor, QQ_I
+    >>> factor(x**2/4 + 1)
+    (x**2 + 4)/4
+    >>> factor(x**2/4 + 1, domain='QQ_I')
+    (x - 2*I)*(x + 2*I)/4
+    >>> factor(x**2/4 + 1, domain=QQ_I)
+    (x - 2*I)*(x + 2*I)/4
+
+    It is also possible to specify the :ref:`QQ_I` domain explicitly with
+    polys functions like :py:func:`~.apart`.
+
+    >>> from sympy import apart
+    >>> apart(1/(1 + x**2))
+    1/(x**2 + 1)
+    >>> apart(1/(1 + x**2), domain=QQ_I)
+    I/(2*(x + I)) - I/(2*(x - I))
+
+    The corresponding `ring of integers`_ is the domain of the Gaussian
+    integers :ref:`ZZ_I`. Conversely :ref:`QQ_I` is the `field of fractions`_
+    of :ref:`ZZ_I`.
+
+    >>> from sympy import ZZ_I, QQ_I, QQ
+    >>> ZZ_I.get_field()
+    QQ_I
+    >>> QQ_I.get_ring()
+    ZZ_I
+
+    When using the domain directly :ref:`QQ_I` can be used as a constructor.
+
+    >>> QQ_I(3, 4)
+    (3 + 4*I)
+    >>> QQ_I(5)
+    (5 + 0*I)
+    >>> QQ_I(QQ(2, 3), QQ(4, 5))
+    (2/3 + 4/5*I)
+
+    The domain elements of :ref:`QQ_I` are instances of
+    :py:class:`~.GaussianRational` which support the field operations
+    ``+,-,*,**,/``.
+
+    >>> z1 = QQ_I(5, 1)
+    >>> z2 = QQ_I(2, QQ(1, 2))
+    >>> z1
+    (5 + 1*I)
+    >>> z2
+    (2 + 1/2*I)
+    >>> z1 + z2
+    (7 + 3/2*I)
+    >>> z1 * z2
+    (19/2 + 9/2*I)
+    >>> z2 ** 2
+    (15/4 + 2*I)
+
+    True division (``/``) in :ref:`QQ_I` gives an element of :ref:`QQ_I` and
+    is always exact.
+
+    >>> z1 / z2
+    (42/17 + -2/17*I)
+    >>> QQ_I.exquo(z1, z2)
+    (42/17 + -2/17*I)
+    >>> z1 == (z1/z2)*z2
+    True
+
+    Both floor (``//``) and modulo (``%``) division can be used with
+    :py:class:`~.GaussianRational` (see :py:meth:`~.Domain.div`)
+    but division is always exact so there is no remainder.
+
+    >>> z1 // z2
+    (42/17 + -2/17*I)
+    >>> z1 % z2
+    (0 + 0*I)
+    >>> QQ_I.div(z1, z2)
+    ((42/17 + -2/17*I), (0 + 0*I))
+    >>> (z1//z2)*z2 + z1%z2 == z1
+    True
+
+    .. _Gaussian rationals: https://en.wikipedia.org/wiki/Gaussian_rational
+    """
+    dom = QQ
+    mod = DMP([QQ.one, QQ.zero, QQ.one], QQ)
+    dtype = GaussianRational
+    zero = dtype(QQ(0), QQ(0))
+    one = dtype(QQ(1), QQ(0))
+    imag_unit = dtype(QQ(0), QQ(1))
+    units = (one, imag_unit, -one, -imag_unit)  # powers of i
+
+    rep = 'QQ_I'
+
+    is_GaussianField = True
+    is_QQ_I = True
+
+    def __init__(self):  # override Domain.__init__
+        """For constructing QQ_I."""
+
+    def __eq__(self, other):
+        """Returns ``True`` if two domains are equivalent. """
+        if isinstance(other, GaussianRationalField):
+            return True
+        else:
+            return NotImplemented
+
+    def __hash__(self):
+        """Compute hash code of ``self``. """
+        return hash('QQ_I')
+
+    @property
+    def has_CharacteristicZero(self):
+        return True
+
+    def characteristic(self):
+        return 0
+
+    def get_ring(self):
+        """Returns a ring associated with ``self``. """
+        return ZZ_I
+
+    def get_field(self):
+        """Returns a field associated with ``self``. """
+        return self
+
+    def as_AlgebraicField(self):
+        """Get equivalent domain as an ``AlgebraicField``. """
+        return AlgebraicField(self.dom, I)
+
+    def numer(self, a):
+        """Get the numerator of ``a``."""
+        ZZ_I = self.get_ring()
+        return ZZ_I.convert(a * self.denom(a))
+
+    def denom(self, a):
+        """Get the denominator of ``a``."""
+        ZZ = self.dom.get_ring()
+        QQ = self.dom
+        ZZ_I = self.get_ring()
+        denom_ZZ = ZZ.lcm(QQ.denom(a.x), QQ.denom(a.y))
+        return ZZ_I(denom_ZZ, ZZ.zero)
+
+    def from_GaussianIntegerRing(K1, a, K0):
+        """Convert a ZZ_I element to QQ_I."""
+        return K1.new(a.x, a.y)
+
+    def from_GaussianRationalField(K1, a, K0):
+        """Convert a QQ_I element to QQ_I."""
+        return a
+
+    def from_ComplexField(K1, a, K0):
+        """Convert a ComplexField element to QQ_I."""
+        return K1.new(QQ.convert(a.real), QQ.convert(a.imag))
+
+
+QQ_I = GaussianRational._parent = GaussianRationalField()

.venv/lib/python3.13/site-packages/sympy/polys/domains/gmpyfinitefield.py

@@ -0,0 +1,16 @@
+"""Implementation of :class:`GMPYFiniteField` class. """
+
+
+from sympy.polys.domains.finitefield import FiniteField
+from sympy.polys.domains.gmpyintegerring import GMPYIntegerRing
+
+from sympy.utilities import public
+
+@public
+class GMPYFiniteField(FiniteField):
+    """Finite field based on GMPY integers. """
+
+    alias = 'FF_gmpy'
+
+    def __init__(self, mod, symmetric=True):
+        super().__init__(mod, GMPYIntegerRing(), symmetric)

.venv/lib/python3.13/site-packages/sympy/polys/domains/gmpyintegerring.py

@@ -0,0 +1,105 @@
+"""Implementation of :class:`GMPYIntegerRing` class. """
+
+
+from sympy.polys.domains.groundtypes import (
+    GMPYInteger, SymPyInteger,
+    factorial as gmpy_factorial,
+    gmpy_gcdex, gmpy_gcd, gmpy_lcm, sqrt as gmpy_sqrt,
+)
+from sympy.core.numbers import int_valued
+from sympy.polys.domains.integerring import IntegerRing
+from sympy.polys.polyerrors import CoercionFailed
+from sympy.utilities import public
+
+@public
+class GMPYIntegerRing(IntegerRing):
+    """Integer ring based on GMPY's ``mpz`` type.
+
+    This will be the implementation of :ref:`ZZ` if ``gmpy`` or ``gmpy2`` is
+    installed. Elements will be of type ``gmpy.mpz``.
+    """
+
+    dtype = GMPYInteger
+    zero = dtype(0)
+    one = dtype(1)
+    tp = type(one)
+    alias = 'ZZ_gmpy'
+
+    def __init__(self):
+        """Allow instantiation of this domain. """
+
+    def to_sympy(self, a):
+        """Convert ``a`` to a SymPy object. """
+        return SymPyInteger(int(a))
+
+    def from_sympy(self, a):
+        """Convert SymPy's Integer to ``dtype``. """
+        if a.is_Integer:
+            return GMPYInteger(a.p)
+        elif int_valued(a):
+            return GMPYInteger(int(a))
+        else:
+            raise CoercionFailed("expected an integer, got %s" % a)
+
+    def from_FF_python(K1, a, K0):
+        """Convert ``ModularInteger(int)`` to GMPY's ``mpz``. """
+        return K0.to_int(a)
+
+    def from_ZZ_python(K1, a, K0):
+        """Convert Python's ``int`` to GMPY's ``mpz``. """
+        return GMPYInteger(a)
+
+    def from_QQ(K1, a, K0):
+        """Convert Python's ``Fraction`` to GMPY's ``mpz``. """
+        if a.denominator == 1:
+            return GMPYInteger(a.numerator)
+
+    def from_QQ_python(K1, a, K0):
+        """Convert Python's ``Fraction`` to GMPY's ``mpz``. """
+        if a.denominator == 1:
+            return GMPYInteger(a.numerator)
+
+    def from_FF_gmpy(K1, a, K0):
+        """Convert ``ModularInteger(mpz)`` to GMPY's ``mpz``. """
+        return K0.to_int(a)
+
+    def from_ZZ_gmpy(K1, a, K0):
+        """Convert GMPY's ``mpz`` to GMPY's ``mpz``. """
+        return a
+
+    def from_QQ_gmpy(K1, a, K0):
+        """Convert GMPY ``mpq`` to GMPY's ``mpz``. """
+        if a.denominator == 1:
+            return a.numerator
+
+    def from_RealField(K1, a, K0):
+        """Convert mpmath's ``mpf`` to GMPY's ``mpz``. """
+        p, q = K0.to_rational(a)
+
+        if q == 1:
+            return GMPYInteger(p)
+
+    def from_GaussianIntegerRing(K1, a, K0):
+        if a.y == 0:
+            return a.x
+
+    def gcdex(self, a, b):
+        """Compute extended GCD of ``a`` and ``b``. """
+        h, s, t = gmpy_gcdex(a, b)
+        return s, t, h
+
+    def gcd(self, a, b):
+        """Compute GCD of ``a`` and ``b``. """
+        return gmpy_gcd(a, b)
+
+    def lcm(self, a, b):
+        """Compute LCM of ``a`` and ``b``. """
+        return gmpy_lcm(a, b)
+
+    def sqrt(self, a):
+        """Compute square root of ``a``. """
+        return gmpy_sqrt(a)
+
+    def factorial(self, a):
+        """Compute factorial of ``a``. """
+        return gmpy_factorial(a)

.venv/lib/python3.13/site-packages/sympy/polys/domains/gmpyrationalfield.py

@@ -0,0 +1,100 @@
+"""Implementation of :class:`GMPYRationalField` class. """
+
+
+from sympy.polys.domains.groundtypes import (
+    GMPYRational, SymPyRational,
+    gmpy_numer, gmpy_denom, factorial as gmpy_factorial,
+)
+from sympy.polys.domains.rationalfield import RationalField
+from sympy.polys.polyerrors import CoercionFailed
+from sympy.utilities import public
+
+@public
+class GMPYRationalField(RationalField):
+    """Rational field based on GMPY's ``mpq`` type.
+
+    This will be the implementation of :ref:`QQ` if ``gmpy`` or ``gmpy2`` is
+    installed. Elements will be of type ``gmpy.mpq``.
+    """
+
+    dtype = GMPYRational
+    zero = dtype(0)
+    one = dtype(1)
+    tp = type(one)
+    alias = 'QQ_gmpy'
+
+    def __init__(self):
+        pass
+
+    def get_ring(self):
+        """Returns ring associated with ``self``. """
+        from sympy.polys.domains import GMPYIntegerRing
+        return GMPYIntegerRing()
+
+    def to_sympy(self, a):
+        """Convert ``a`` to a SymPy object. """
+        return SymPyRational(int(gmpy_numer(a)),
+                             int(gmpy_denom(a)))
+
+    def from_sympy(self, a):
+        """Convert SymPy's Integer to ``dtype``. """
+        if a.is_Rational:
+            return GMPYRational(a.p, a.q)
+        elif a.is_Float:
+            from sympy.polys.domains import RR
+            return GMPYRational(*map(int, RR.to_rational(a)))
+        else:
+            raise CoercionFailed("expected ``Rational`` object, got %s" % a)
+
+    def from_ZZ_python(K1, a, K0):
+        """Convert a Python ``int`` object to ``dtype``. """
+        return GMPYRational(a)
+
+    def from_QQ_python(K1, a, K0):
+        """Convert a Python ``Fraction`` object to ``dtype``. """
+        return GMPYRational(a.numerator, a.denominator)
+
+    def from_ZZ_gmpy(K1, a, K0):
+        """Convert a GMPY ``mpz`` object to ``dtype``. """
+        return GMPYRational(a)
+
+    def from_QQ_gmpy(K1, a, K0):
+        """Convert a GMPY ``mpq`` object to ``dtype``. """
+        return a
+
+    def from_GaussianRationalField(K1, a, K0):
+        """Convert a ``GaussianElement`` object to ``dtype``. """
+        if a.y == 0:
+            return GMPYRational(a.x)
+
+    def from_RealField(K1, a, K0):
+        """Convert a mpmath ``mpf`` object to ``dtype``. """
+        return GMPYRational(*map(int, K0.to_rational(a)))
+
+    def exquo(self, a, b):
+        """Exact quotient of ``a`` and ``b``, implies ``__truediv__``.  """
+        return GMPYRational(a) / GMPYRational(b)
+
+    def quo(self, a, b):
+        """Quotient of ``a`` and ``b``, implies ``__truediv__``. """
+        return GMPYRational(a) / GMPYRational(b)
+
+    def rem(self, a, b):
+        """Remainder of ``a`` and ``b``, implies nothing.  """
+        return self.zero
+
+    def div(self, a, b):
+        """Division of ``a`` and ``b``, implies ``__truediv__``. """
+        return GMPYRational(a) / GMPYRational(b), self.zero
+
+    def numer(self, a):
+        """Returns numerator of ``a``. """
+        return a.numerator
+
+    def denom(self, a):
+        """Returns denominator of ``a``. """
+        return a.denominator
+
+    def factorial(self, a):
+        """Returns factorial of ``a``. """
+        return GMPYRational(gmpy_factorial(int(a)))

.venv/lib/python3.13/site-packages/sympy/polys/domains/groundtypes.py

@@ -0,0 +1,99 @@
+"""Ground types for various mathematical domains in SymPy. """
+
+import builtins
+from sympy.external.gmpy import GROUND_TYPES, factorial, sqrt, is_square, sqrtrem
+
+PythonInteger = builtins.int
+PythonReal = builtins.float
+PythonComplex = builtins.complex
+
+from .pythonrational import PythonRational
+
+from sympy.core.intfunc import (
+    igcdex as python_gcdex,
+    igcd2 as python_gcd,
+    ilcm as python_lcm,
+)
+
+from sympy.core.numbers import (Float as SymPyReal, Integer as SymPyInteger, Rational as SymPyRational)
+
+
+class _GMPYInteger:
+    def __init__(self, obj):
+        pass
+
+class _GMPYRational:
+    def __init__(self, obj):
+        pass
+
+
+if GROUND_TYPES == 'gmpy':
+
+    from gmpy2 import (
+        mpz as GMPYInteger,
+        mpq as GMPYRational,
+        numer as gmpy_numer,
+        denom as gmpy_denom,
+        gcdext as gmpy_gcdex,
+        gcd as gmpy_gcd,
+        lcm as gmpy_lcm,
+        qdiv as gmpy_qdiv,
+    )
+    gcdex = gmpy_gcdex
+    gcd = gmpy_gcd
+    lcm = gmpy_lcm
+
+elif GROUND_TYPES == 'flint':
+
+    from flint import fmpz as _fmpz
+
+    GMPYInteger = _GMPYInteger
+    GMPYRational = _GMPYRational
+    gmpy_numer = None
+    gmpy_denom = None
+    gmpy_gcdex = None
+    gmpy_gcd = None
+    gmpy_lcm = None
+    gmpy_qdiv = None
+
+    def gcd(a, b):
+        return a.gcd(b)
+
+    def gcdex(a, b):
+        x, y, g = python_gcdex(a, b)
+        return _fmpz(x), _fmpz(y), _fmpz(g)
+
+    def lcm(a, b):
+        return a.lcm(b)
+
+else:
+    GMPYInteger = _GMPYInteger
+    GMPYRational = _GMPYRational
+    gmpy_numer = None
+    gmpy_denom = None
+    gmpy_gcdex = None
+    gmpy_gcd = None
+    gmpy_lcm = None
+    gmpy_qdiv = None
+    gcdex = python_gcdex
+    gcd = python_gcd
+    lcm = python_lcm
+
+
+__all__ = [
+    'PythonInteger', 'PythonReal', 'PythonComplex',
+
+    'PythonRational',
+
+    'python_gcdex', 'python_gcd', 'python_lcm',
+
+    'SymPyReal', 'SymPyInteger', 'SymPyRational',
+
+    'GMPYInteger', 'GMPYRational', 'gmpy_numer',
+    'gmpy_denom', 'gmpy_gcdex', 'gmpy_gcd', 'gmpy_lcm',
+    'gmpy_qdiv',
+
+    'factorial', 'sqrt', 'is_square', 'sqrtrem',
+
+    'GMPYInteger', 'GMPYRational',
+]

.venv/lib/python3.13/site-packages/sympy/polys/domains/integerring.py

@@ -0,0 +1,276 @@
+"""Implementation of :class:`IntegerRing` class. """
+
+from sympy.external.gmpy import MPZ, GROUND_TYPES
+
+from sympy.core.numbers import int_valued
+from sympy.polys.domains.groundtypes import (
+    SymPyInteger,
+    factorial,
+    gcdex, gcd, lcm, sqrt, is_square, sqrtrem,
+)
+
+from sympy.polys.domains.characteristiczero import CharacteristicZero
+from sympy.polys.domains.ring import Ring
+from sympy.polys.domains.simpledomain import SimpleDomain
+from sympy.polys.polyerrors import CoercionFailed
+from sympy.utilities import public
+
+import math
+
+@public
+class IntegerRing(Ring, CharacteristicZero, SimpleDomain):
+    r"""The domain ``ZZ`` representing the integers `\mathbb{Z}`.
+
+    The :py:class:`IntegerRing` class represents the ring of integers as a
+    :py:class:`~.Domain` in the domain system. :py:class:`IntegerRing` is a
+    super class of :py:class:`PythonIntegerRing` and
+    :py:class:`GMPYIntegerRing` one of which will be the implementation for
+    :ref:`ZZ` depending on whether or not ``gmpy`` or ``gmpy2`` is installed.
+
+    See also
+    ========
+
+    Domain
+    """
+
+    rep = 'ZZ'
+    alias = 'ZZ'
+    dtype = MPZ
+    zero = dtype(0)
+    one = dtype(1)
+    tp = type(one)
+
+
+    is_IntegerRing = is_ZZ = True
+    is_Numerical = True
+    is_PID = True
+
+    has_assoc_Ring = True
+    has_assoc_Field = True
+
+    def __init__(self):
+        """Allow instantiation of this domain. """
+
+    def __eq__(self, other):
+        """Returns ``True`` if two domains are equivalent. """
+        if isinstance(other, IntegerRing):
+            return True
+        else:
+            return NotImplemented
+
+    def __hash__(self):
+        """Compute a hash value for this domain. """
+        return hash('ZZ')
+
+    def to_sympy(self, a):
+        """Convert ``a`` to a SymPy object. """
+        return SymPyInteger(int(a))
+
+    def from_sympy(self, a):
+        """Convert SymPy's Integer to ``dtype``. """
+        if a.is_Integer:
+            return MPZ(a.p)
+        elif int_valued(a):
+            return MPZ(int(a))
+        else:
+            raise CoercionFailed("expected an integer, got %s" % a)
+
+    def get_field(self):
+        r"""Return the associated field of fractions :ref:`QQ`
+
+        Returns
+        =======
+
+        :ref:`QQ`:
+            The associated field of fractions :ref:`QQ`, a
+            :py:class:`~.Domain` representing the rational numbers
+            `\mathbb{Q}`.
+
+        Examples
+        ========
+
+        >>> from sympy import ZZ
+        >>> ZZ.get_field()
+        QQ
+        """
+        from sympy.polys.domains import QQ
+        return QQ
+
+    def algebraic_field(self, *extension, alias=None):
+        r"""Returns an algebraic field, i.e. `\mathbb{Q}(\alpha, \ldots)`.
+
+        Parameters
+        ==========
+
+        *extension : One or more :py:class:`~.Expr`.
+            Generators of the extension. These should be expressions that are
+            algebraic over `\mathbb{Q}`.
+
+        alias : str, :py:class:`~.Symbol`, None, optional (default=None)
+            If provided, this will be used as the alias symbol for the
+            primitive element of the returned :py:class:`~.AlgebraicField`.
+
+        Returns
+        =======
+
+        :py:class:`~.AlgebraicField`
+            A :py:class:`~.Domain` representing the algebraic field extension.
+
+        Examples
+        ========
+
+        >>> from sympy import ZZ, sqrt
+        >>> ZZ.algebraic_field(sqrt(2))
+        QQ<sqrt(2)>
+        """
+        return self.get_field().algebraic_field(*extension, alias=alias)
+
+    def from_AlgebraicField(K1, a, K0):
+        """Convert a :py:class:`~.ANP` object to :ref:`ZZ`.
+
+        See :py:meth:`~.Domain.convert`.
+        """
+        if a.is_ground:
+            return K1.convert(a.LC(), K0.dom)
+
+    def log(self, a, b):
+        r"""Logarithm of *a* to the base *b*.
+
+        Parameters
+        ==========
+
+        a: number
+        b: number
+
+        Returns
+        =======
+
+        $\\lfloor\log(a, b)\\rfloor$:
+            Floor of the logarithm of *a* to the base *b*
+
+        Examples
+        ========
+
+        >>> from sympy import ZZ
+        >>> ZZ.log(ZZ(8), ZZ(2))
+        3
+        >>> ZZ.log(ZZ(9), ZZ(2))
+        3
+
+        Notes
+        =====
+
+        This function uses ``math.log`` which is based on ``float`` so it will
+        fail for large integer arguments.
+        """
+        return self.dtype(int(math.log(int(a), b)))
+
+    def from_FF(K1, a, K0):
+        """Convert ``ModularInteger(int)`` to GMPY's ``mpz``. """
+        return MPZ(K0.to_int(a))
+
+    def from_FF_python(K1, a, K0):
+        """Convert ``ModularInteger(int)`` to GMPY's ``mpz``. """
+        return MPZ(K0.to_int(a))
+
+    def from_ZZ(K1, a, K0):
+        """Convert Python's ``int`` to GMPY's ``mpz``. """
+        return MPZ(a)
+
+    def from_ZZ_python(K1, a, K0):
+        """Convert Python's ``int`` to GMPY's ``mpz``. """
+        return MPZ(a)
+
+    def from_QQ(K1, a, K0):
+        """Convert Python's ``Fraction`` to GMPY's ``mpz``. """
+        if a.denominator == 1:
+            return MPZ(a.numerator)
+
+    def from_QQ_python(K1, a, K0):
+        """Convert Python's ``Fraction`` to GMPY's ``mpz``. """
+        if a.denominator == 1:
+            return MPZ(a.numerator)
+
+    def from_FF_gmpy(K1, a, K0):
+        """Convert ``ModularInteger(mpz)`` to GMPY's ``mpz``. """
+        return MPZ(K0.to_int(a))
+
+    def from_ZZ_gmpy(K1, a, K0):
+        """Convert GMPY's ``mpz`` to GMPY's ``mpz``. """
+        return a
+
+    def from_QQ_gmpy(K1, a, K0):
+        """Convert GMPY ``mpq`` to GMPY's ``mpz``. """
+        if a.denominator == 1:
+            return a.numerator
+
+    def from_RealField(K1, a, K0):
+        """Convert mpmath's ``mpf`` to GMPY's ``mpz``. """
+        p, q = K0.to_rational(a)
+
+        if q == 1:
+            # XXX: If MPZ is flint.fmpz and p is a gmpy2.mpz, then we need
+            # to convert via int because fmpz and mpz do not know about each
+            # other.
+            return MPZ(int(p))
+
+    def from_GaussianIntegerRing(K1, a, K0):
+        if a.y == 0:
+            return a.x
+
+    def from_EX(K1, a, K0):
+        """Convert ``Expression`` to GMPY's ``mpz``. """
+        if a.is_Integer:
+            return K1.from_sympy(a)
+
+    def gcdex(self, a, b):
+        """Compute extended GCD of ``a`` and ``b``. """
+        h, s, t = gcdex(a, b)
+        # XXX: This conditional logic should be handled somewhere else.
+        if GROUND_TYPES == 'gmpy':
+            return s, t, h
+        else:
+            return h, s, t
+
+    def gcd(self, a, b):
+        """Compute GCD of ``a`` and ``b``. """
+        return gcd(a, b)
+
+    def lcm(self, a, b):
+        """Compute LCM of ``a`` and ``b``. """
+        return lcm(a, b)
+
+    def sqrt(self, a):
+        """Compute square root of ``a``. """
+        return sqrt(a)
+
+    def is_square(self, a):
+        """Return ``True`` if ``a`` is a square.
+
+        Explanation
+        ===========
+        An integer is a square if and only if there exists an integer
+        ``b`` such that ``b * b == a``.
+        """
+        return is_square(a)
+
+    def exsqrt(self, a):
+        """Non-negative square root of ``a`` if ``a`` is a square.
+
+        See also
+        ========
+        is_square
+        """
+        if a < 0:
+            return None
+        root, rem = sqrtrem(a)
+        if rem != 0:
+            return None
+        return root
+
+    def factorial(self, a):
+        """Compute factorial of ``a``. """
+        return factorial(a)
+
+
+ZZ = IntegerRing()

.venv/lib/python3.13/site-packages/sympy/polys/domains/modularinteger.py

@@ -0,0 +1,237 @@
+"""Implementation of :class:`ModularInteger` class. """
+
+from __future__ import annotations
+from typing import Any
+
+import operator
+
+from sympy.polys.polyutils import PicklableWithSlots
+from sympy.polys.polyerrors import CoercionFailed
+from sympy.polys.domains.domainelement import DomainElement
+
+from sympy.utilities import public
+from sympy.utilities.exceptions import sympy_deprecation_warning
+
+@public
+class ModularInteger(PicklableWithSlots, DomainElement):
+    """A class representing a modular integer. """
+
+    mod, dom, sym, _parent = None, None, None, None
+
+    __slots__ = ('val',)
+
+    def parent(self):
+        return self._parent
+
+    def __init__(self, val):
+        if isinstance(val, self.__class__):
+            self.val = val.val % self.mod
+        else:
+            self.val = self.dom.convert(val) % self.mod
+
+    def modulus(self):
+        return self.mod
+
+    def __hash__(self):
+        return hash((self.val, self.mod))
+
+    def __repr__(self):
+        return "%s(%s)" % (self.__class__.__name__, self.val)
+
+    def __str__(self):
+        return "%s mod %s" % (self.val, self.mod)
+
+    def __int__(self):
+        return int(self.val)
+
+    def to_int(self):
+
+        sympy_deprecation_warning(
+            """ModularInteger.to_int() is deprecated.
+
+            Use int(a) or K = GF(p) and K.to_int(a) instead of a.to_int().
+            """,
+            deprecated_since_version="1.13",
+            active_deprecations_target="modularinteger-to-int",
+        )
+
+        if self.sym:
+            if self.val <= self.mod // 2:
+                return self.val
+            else:
+                return self.val - self.mod
+        else:
+            return self.val
+
+    def __pos__(self):
+        return self
+
+    def __neg__(self):
+        return self.__class__(-self.val)
+
+    @classmethod
+    def _get_val(cls, other):
+        if isinstance(other, cls):
+            return other.val
+        else:
+            try:
+                return cls.dom.convert(other)
+            except CoercionFailed:
+                return None
+
+    def __add__(self, other):
+        val = self._get_val(other)
+
+        if val is not None:
+            return self.__class__(self.val + val)
+        else:
+            return NotImplemented
+
+    def __radd__(self, other):
+        return self.__add__(other)
+
+    def __sub__(self, other):
+        val = self._get_val(other)
+
+        if val is not None:
+            return self.__class__(self.val - val)
+        else:
+            return NotImplemented
+
+    def __rsub__(self, other):
+        return (-self).__add__(other)
+
+    def __mul__(self, other):
+        val = self._get_val(other)
+
+        if val is not None:
+            return self.__class__(self.val * val)
+        else:
+            return NotImplemented
+
+    def __rmul__(self, other):
+        return self.__mul__(other)
+
+    def __truediv__(self, other):
+        val = self._get_val(other)
+
+        if val is not None:
+            return self.__class__(self.val * self._invert(val))
+        else:
+            return NotImplemented
+
+    def __rtruediv__(self, other):
+        return self.invert().__mul__(other)
+
+    def __mod__(self, other):
+        val = self._get_val(other)
+
+        if val is not None:
+            return self.__class__(self.val % val)
+        else:
+            return NotImplemented
+
+    def __rmod__(self, other):
+        val = self._get_val(other)
+
+        if val is not None:
+            return self.__class__(val % self.val)
+        else:
+            return NotImplemented
+
+    def __pow__(self, exp):
+        if not exp:
+            return self.__class__(self.dom.one)
+
+        if exp < 0:
+            val, exp = self.invert().val, -exp
+        else:
+            val = self.val
+
+        return self.__class__(pow(val, int(exp), self.mod))
+
+    def _compare(self, other, op):
+        val = self._get_val(other)
+
+        if val is None:
+            return NotImplemented
+
+        return op(self.val, val % self.mod)
+
+    def _compare_deprecated(self, other, op):
+        val = self._get_val(other)
+
+        if val is None:
+            return NotImplemented
+
+        sympy_deprecation_warning(
+            """Ordered comparisons with modular integers are deprecated.
+
+            Use e.g. int(a) < int(b) instead of a < b.
+            """,
+            deprecated_since_version="1.13",
+            active_deprecations_target="modularinteger-compare",
+            stacklevel=4,
+        )
+
+        return op(self.val, val % self.mod)
+
+    def __eq__(self, other):
+        return self._compare(other, operator.eq)
+
+    def __ne__(self, other):
+        return self._compare(other, operator.ne)
+
+    def __lt__(self, other):
+        return self._compare_deprecated(other, operator.lt)
+
+    def __le__(self, other):
+        return self._compare_deprecated(other, operator.le)
+
+    def __gt__(self, other):
+        return self._compare_deprecated(other, operator.gt)
+
+    def __ge__(self, other):
+        return self._compare_deprecated(other, operator.ge)
+
+    def __bool__(self):
+        return bool(self.val)
+
+    @classmethod
+    def _invert(cls, value):
+        return cls.dom.invert(value, cls.mod)
+
+    def invert(self):
+        return self.__class__(self._invert(self.val))
+
+_modular_integer_cache: dict[tuple[Any, Any, Any], type[ModularInteger]] = {}
+
+def ModularIntegerFactory(_mod, _dom, _sym, parent):
+    """Create custom class for specific integer modulus."""
+    try:
+        _mod = _dom.convert(_mod)
+    except CoercionFailed:
+        ok = False
+    else:
+        ok = True
+
+    if not ok or _mod < 1:
+        raise ValueError("modulus must be a positive integer, got %s" % _mod)
+
+    key = _mod, _dom, _sym
+
+    try:
+        cls = _modular_integer_cache[key]
+    except KeyError:
+        class cls(ModularInteger):
+            mod, dom, sym = _mod, _dom, _sym
+            _parent = parent
+
+        if _sym:
+            cls.__name__ = "SymmetricModularIntegerMod%s" % _mod
+        else:
+            cls.__name__ = "ModularIntegerMod%s" % _mod
+
+        _modular_integer_cache[key] = cls
+
+    return cls

.venv/lib/python3.13/site-packages/sympy/polys/domains/mpelements.py

@@ -0,0 +1,181 @@
+#
+# This module is deprecated and should not be used any more. The actual
+# implementation of RR and CC now uses mpmath's mpf and mpc types directly.
+#
+"""Real and complex elements. """
+
+
+from sympy.external.gmpy import MPQ
+from sympy.polys.domains.domainelement import DomainElement
+from sympy.utilities import public
+
+from mpmath.ctx_mp_python import PythonMPContext, _mpf, _mpc, _constant
+from mpmath.libmp import (MPZ_ONE, fzero, fone, finf, fninf, fnan,
+    round_nearest, mpf_mul, repr_dps, int_types,
+    from_int, from_float, from_str, to_rational)
+
+
+@public
+class RealElement(_mpf, DomainElement):
+    """An element of a real domain. """
+
+    __slots__ = ('__mpf__',)
+
+    def _set_mpf(self, val):
+        self.__mpf__ = val
+
+    _mpf_ = property(lambda self: self.__mpf__, _set_mpf)
+
+    def parent(self):
+        return self.context._parent
+
+@public
+class ComplexElement(_mpc, DomainElement):
+    """An element of a complex domain. """
+
+    __slots__ = ('__mpc__',)
+
+    def _set_mpc(self, val):
+        self.__mpc__ = val
+
+    _mpc_ = property(lambda self: self.__mpc__, _set_mpc)
+
+    def parent(self):
+        return self.context._parent
+
+new = object.__new__
+
+@public
+class MPContext(PythonMPContext):
+
+    def __init__(ctx, prec=53, dps=None, tol=None, real=False):
+        ctx._prec_rounding = [prec, round_nearest]
+
+        if dps is None:
+            ctx._set_prec(prec)
+        else:
+            ctx._set_dps(dps)
+
+        ctx.mpf = RealElement
+        ctx.mpc = ComplexElement
+        ctx.mpf._ctxdata = [ctx.mpf, new, ctx._prec_rounding]
+        ctx.mpc._ctxdata = [ctx.mpc, new, ctx._prec_rounding]
+
+        if real:
+            ctx.mpf.context = ctx
+        else:
+            ctx.mpc.context = ctx
+
+        ctx.constant = _constant
+        ctx.constant._ctxdata = [ctx.mpf, new, ctx._prec_rounding]
+        ctx.constant.context = ctx
+
+        ctx.types = [ctx.mpf, ctx.mpc, ctx.constant]
+        ctx.trap_complex = True
+        ctx.pretty = True
+
+        if tol is None:
+            ctx.tol = ctx._make_tol()
+        elif tol is False:
+            ctx.tol = fzero
+        else:
+            ctx.tol = ctx._convert_tol(tol)
+
+        ctx.tolerance = ctx.make_mpf(ctx.tol)
+
+        if not ctx.tolerance:
+            ctx.max_denom = 1000000
+        else:
+            ctx.max_denom = int(1/ctx.tolerance)
+
+        ctx.zero = ctx.make_mpf(fzero)
+        ctx.one = ctx.make_mpf(fone)
+        ctx.j = ctx.make_mpc((fzero, fone))
+        ctx.inf = ctx.make_mpf(finf)
+        ctx.ninf = ctx.make_mpf(fninf)
+        ctx.nan = ctx.make_mpf(fnan)
+
+    def _make_tol(ctx):
+        hundred = (0, 25, 2, 5)
+        eps = (0, MPZ_ONE, 1-ctx.prec, 1)
+        return mpf_mul(hundred, eps)
+
+    def make_tol(ctx):
+        return ctx.make_mpf(ctx._make_tol())
+
+    def _convert_tol(ctx, tol):
+        if isinstance(tol, int_types):
+            return from_int(tol)
+        if isinstance(tol, float):
+            return from_float(tol)
+        if hasattr(tol, "_mpf_"):
+            return tol._mpf_
+        prec, rounding = ctx._prec_rounding
+        if isinstance(tol, str):
+            return from_str(tol, prec, rounding)
+        raise ValueError("expected a real number, got %s" % tol)
+
+    def _convert_fallback(ctx, x, strings):
+        raise TypeError("cannot create mpf from " + repr(x))
+
+    @property
+    def _repr_digits(ctx):
+        return repr_dps(ctx._prec)
+
+    @property
+    def _str_digits(ctx):
+        return ctx._dps
+
+    def to_rational(ctx, s, limit=True):
+        p, q = to_rational(s._mpf_)
+
+        # Needed for GROUND_TYPES=flint if gmpy2 is installed because mpmath's
+        # to_rational() function returns a gmpy2.mpz instance and if MPQ is
+        # flint.fmpq then MPQ(p, q) will fail.
+        p = int(p)
+
+        if not limit or q <= ctx.max_denom:
+            return p, q
+
+        p0, q0, p1, q1 = 0, 1, 1, 0
+        n, d = p, q
+
+        while True:
+            a = n//d
+            q2 = q0 + a*q1
+            if q2 > ctx.max_denom:
+                break
+            p0, q0, p1, q1 = p1, q1, p0 + a*p1, q2
+            n, d = d, n - a*d
+
+        k = (ctx.max_denom - q0)//q1
+
+        number = MPQ(p, q)
+        bound1 = MPQ(p0 + k*p1, q0 + k*q1)
+        bound2 = MPQ(p1, q1)
+
+        if not bound2 or not bound1:
+            return p, q
+        elif abs(bound2 - number) <= abs(bound1 - number):
+            return bound2.numerator, bound2.denominator
+        else:
+            return bound1.numerator, bound1.denominator
+
+    def almosteq(ctx, s, t, rel_eps=None, abs_eps=None):
+        t = ctx.convert(t)
+        if abs_eps is None and rel_eps is None:
+            rel_eps = abs_eps = ctx.tolerance or ctx.make_tol()
+        if abs_eps is None:
+            abs_eps = ctx.convert(rel_eps)
+        elif rel_eps is None:
+            rel_eps = ctx.convert(abs_eps)
+        diff = abs(s-t)
+        if diff <= abs_eps:
+            return True
+        abss = abs(s)
+        abst = abs(t)
+        if abss < abst:
+            err = diff/abst
+        else:
+            err = diff/abss
+        return err <= rel_eps

.venv/lib/python3.13/site-packages/sympy/polys/domains/old_fractionfield.py

@@ -0,0 +1,188 @@
+"""Implementation of :class:`FractionField` class. """
+
+
+from sympy.polys.domains.field import Field
+from sympy.polys.domains.compositedomain import CompositeDomain
+from sympy.polys.polyclasses import DMF
+from sympy.polys.polyerrors import GeneratorsNeeded
+from sympy.polys.polyutils import dict_from_basic, basic_from_dict, _dict_reorder
+from sympy.utilities import public
+
+@public
+class FractionField(Field, CompositeDomain):
+    """A class for representing rational function fields. """
+
+    dtype = DMF
+    is_FractionField = is_Frac = True
+
+    has_assoc_Ring = True
+    has_assoc_Field = True
+
+    def __init__(self, dom, *gens):
+        if not gens:
+            raise GeneratorsNeeded("generators not specified")
+
+        lev = len(gens) - 1
+        self.ngens = len(gens)
+
+        self.zero = self.dtype.zero(lev, dom)
+        self.one = self.dtype.one(lev, dom)
+
+        self.domain = self.dom = dom
+        self.symbols = self.gens = gens
+
+    def set_domain(self, dom):
+        """Make a new fraction field with given domain. """
+        return self.__class__(dom, *self.gens)
+
+    def new(self, element):
+        return self.dtype(element, self.dom, len(self.gens) - 1)
+
+    def __str__(self):
+        return str(self.dom) + '(' + ','.join(map(str, self.gens)) + ')'
+
+    def __hash__(self):
+        return hash((self.__class__.__name__, self.dtype, self.dom, self.gens))
+
+    def __eq__(self, other):
+        """Returns ``True`` if two domains are equivalent. """
+        return isinstance(other, FractionField) and \
+            self.dtype == other.dtype and self.dom == other.dom and self.gens == other.gens
+
+    def to_sympy(self, a):
+        """Convert ``a`` to a SymPy object. """
+        return (basic_from_dict(a.numer().to_sympy_dict(), *self.gens) /
+                basic_from_dict(a.denom().to_sympy_dict(), *self.gens))
+
+    def from_sympy(self, a):
+        """Convert SymPy's expression to ``dtype``. """
+        p, q = a.as_numer_denom()
+
+        num, _ = dict_from_basic(p, gens=self.gens)
+        den, _ = dict_from_basic(q, gens=self.gens)
+
+        for k, v in num.items():
+            num[k] = self.dom.from_sympy(v)
+
+        for k, v in den.items():
+            den[k] = self.dom.from_sympy(v)
+
+        return self((num, den)).cancel()
+
+    def from_ZZ(K1, a, K0):
+        """Convert a Python ``int`` object to ``dtype``. """
+        return K1(K1.dom.convert(a, K0))
+
+    def from_ZZ_python(K1, a, K0):
+        """Convert a Python ``int`` object to ``dtype``. """
+        return K1(K1.dom.convert(a, K0))
+
+    def from_QQ_python(K1, a, K0):
+        """Convert a Python ``Fraction`` object to ``dtype``. """
+        return K1(K1.dom.convert(a, K0))
+
+    def from_ZZ_gmpy(K1, a, K0):
+        """Convert a GMPY ``mpz`` object to ``dtype``. """
+        return K1(K1.dom.convert(a, K0))
+
+    def from_QQ_gmpy(K1, a, K0):
+        """Convert a GMPY ``mpq`` object to ``dtype``. """
+        return K1(K1.dom.convert(a, K0))
+
+    def from_RealField(K1, a, K0):
+        """Convert a mpmath ``mpf`` object to ``dtype``. """
+        return K1(K1.dom.convert(a, K0))
+
+    def from_GlobalPolynomialRing(K1, a, K0):
+        """Convert a ``DMF`` object to ``dtype``. """
+        if K1.gens == K0.gens:
+            if K1.dom == K0.dom:
+                return K1(a.to_list())
+            else:
+                return K1(a.convert(K1.dom).to_list())
+        else:
+            monoms, coeffs = _dict_reorder(a.to_dict(), K0.gens, K1.gens)
+
+            if K1.dom != K0.dom:
+                coeffs = [ K1.dom.convert(c, K0.dom) for c in coeffs ]
+
+            return K1(dict(zip(monoms, coeffs)))
+
+    def from_FractionField(K1, a, K0):
+        """
+        Convert a fraction field element to another fraction field.
+
+        Examples
+        ========
+
+        >>> from sympy.polys.polyclasses import DMF
+        >>> from sympy.polys.domains import ZZ, QQ
+        >>> from sympy.abc import x
+
+        >>> f = DMF(([ZZ(1), ZZ(2)], [ZZ(1), ZZ(1)]), ZZ)
+
+        >>> QQx = QQ.old_frac_field(x)
+        >>> ZZx = ZZ.old_frac_field(x)
+
+        >>> QQx.from_FractionField(f, ZZx)
+        DMF([1, 2], [1, 1], QQ)
+
+        """
+        if K1.gens == K0.gens:
+            if K1.dom == K0.dom:
+                return a
+            else:
+                return K1((a.numer().convert(K1.dom).to_list(),
+                           a.denom().convert(K1.dom).to_list()))
+        elif set(K0.gens).issubset(K1.gens):
+            nmonoms, ncoeffs = _dict_reorder(
+                a.numer().to_dict(), K0.gens, K1.gens)
+            dmonoms, dcoeffs = _dict_reorder(
+                a.denom().to_dict(), K0.gens, K1.gens)
+
+            if K1.dom != K0.dom:
+                ncoeffs = [ K1.dom.convert(c, K0.dom) for c in ncoeffs ]
+                dcoeffs = [ K1.dom.convert(c, K0.dom) for c in dcoeffs ]
+
+            return K1((dict(zip(nmonoms, ncoeffs)), dict(zip(dmonoms, dcoeffs))))
+
+    def get_ring(self):
+        """Returns a ring associated with ``self``. """
+        from sympy.polys.domains import PolynomialRing
+        return PolynomialRing(self.dom, *self.gens)
+
+    def poly_ring(self, *gens):
+        """Returns a polynomial ring, i.e. `K[X]`. """
+        raise NotImplementedError('nested domains not allowed')
+
+    def frac_field(self, *gens):
+        """Returns a fraction field, i.e. `K(X)`. """
+        raise NotImplementedError('nested domains not allowed')
+
+    def is_positive(self, a):
+        """Returns True if ``a`` is positive. """
+        return self.dom.is_positive(a.numer().LC())
+
+    def is_negative(self, a):
+        """Returns True if ``a`` is negative. """
+        return self.dom.is_negative(a.numer().LC())
+
+    def is_nonpositive(self, a):
+        """Returns True if ``a`` is non-positive. """
+        return self.dom.is_nonpositive(a.numer().LC())
+
+    def is_nonnegative(self, a):
+        """Returns True if ``a`` is non-negative. """
+        return self.dom.is_nonnegative(a.numer().LC())
+
+    def numer(self, a):
+        """Returns numerator of ``a``. """
+        return a.numer()
+
+    def denom(self, a):
+        """Returns denominator of ``a``. """
+        return a.denom()
+
+    def factorial(self, a):
+        """Returns factorial of ``a``. """
+        return self.dtype(self.dom.factorial(a))

.venv/lib/python3.13/site-packages/sympy/polys/domains/old_polynomialring.py

@@ -0,0 +1,490 @@
+"""Implementation of :class:`PolynomialRing` class. """
+
+
+from sympy.polys.agca.modules import FreeModulePolyRing
+from sympy.polys.domains.compositedomain import CompositeDomain
+from sympy.polys.domains.old_fractionfield import FractionField
+from sympy.polys.domains.ring import Ring
+from sympy.polys.orderings import monomial_key, build_product_order
+from sympy.polys.polyclasses import DMP, DMF
+from sympy.polys.polyerrors import (GeneratorsNeeded, PolynomialError,
+        CoercionFailed, ExactQuotientFailed, NotReversible)
+from sympy.polys.polyutils import dict_from_basic, basic_from_dict, _dict_reorder
+from sympy.utilities import public
+from sympy.utilities.iterables import iterable
+
+
+@public
+class PolynomialRingBase(Ring, CompositeDomain):
+    """
+    Base class for generalized polynomial rings.
+
+    This base class should be used for uniform access to generalized polynomial
+    rings. Subclasses only supply information about the element storage etc.
+
+    Do not instantiate.
+    """
+
+    has_assoc_Ring = True
+    has_assoc_Field = True
+
+    default_order = "grevlex"
+
+    def __init__(self, dom, *gens, **opts):
+        if not gens:
+            raise GeneratorsNeeded("generators not specified")
+
+        lev = len(gens) - 1
+        self.ngens = len(gens)
+
+        self.zero = self.dtype.zero(lev, dom)
+        self.one = self.dtype.one(lev, dom)
+
+        self.domain = self.dom = dom
+        self.symbols = self.gens = gens
+        # NOTE 'order' may not be set if inject was called through CompositeDomain
+        self.order = opts.get('order', monomial_key(self.default_order))
+
+    def set_domain(self, dom):
+        """Return a new polynomial ring with given domain. """
+        return self.__class__(dom, *self.gens, order=self.order)
+
+    def new(self, element):
+        return self.dtype(element, self.dom, len(self.gens) - 1)
+
+    def _ground_new(self, element):
+        return self.one.ground_new(element)
+
+    def _from_dict(self, element):
+        return DMP.from_dict(element, len(self.gens) - 1, self.dom)
+
+    def __str__(self):
+        s_order = str(self.order)
+        orderstr = (
+            " order=" + s_order) if s_order != self.default_order else ""
+        return str(self.dom) + '[' + ','.join(map(str, self.gens)) + orderstr + ']'
+
+    def __hash__(self):
+        return hash((self.__class__.__name__, self.dtype, self.dom,
+                     self.gens, self.order))
+
+    def __eq__(self, other):
+        """Returns ``True`` if two domains are equivalent. """
+        return isinstance(other, PolynomialRingBase) and \
+            self.dtype == other.dtype and self.dom == other.dom and \
+            self.gens == other.gens and self.order == other.order
+
+    def from_ZZ(K1, a, K0):
+        """Convert a Python ``int`` object to ``dtype``. """
+        return K1._ground_new(K1.dom.convert(a, K0))
+
+    def from_ZZ_python(K1, a, K0):
+        """Convert a Python ``int`` object to ``dtype``. """
+        return K1._ground_new(K1.dom.convert(a, K0))
+
+    def from_QQ(K1, a, K0):
+        """Convert a Python ``Fraction`` object to ``dtype``. """
+        return K1._ground_new(K1.dom.convert(a, K0))
+
+    def from_QQ_python(K1, a, K0):
+        """Convert a Python ``Fraction`` object to ``dtype``. """
+        return K1._ground_new(K1.dom.convert(a, K0))
+
+    def from_ZZ_gmpy(K1, a, K0):
+        """Convert a GMPY ``mpz`` object to ``dtype``. """
+        return K1._ground_new(K1.dom.convert(a, K0))
+
+    def from_QQ_gmpy(K1, a, K0):
+        """Convert a GMPY ``mpq`` object to ``dtype``. """
+        return K1._ground_new(K1.dom.convert(a, K0))
+
+    def from_RealField(K1, a, K0):
+        """Convert a mpmath ``mpf`` object to ``dtype``. """
+        return K1._ground_new(K1.dom.convert(a, K0))
+
+    def from_AlgebraicField(K1, a, K0):
+        """Convert a ``ANP`` object to ``dtype``. """
+        if K1.dom == K0:
+            return K1._ground_new(a)
+
+    def from_PolynomialRing(K1, a, K0):
+        """Convert a ``PolyElement`` object to ``dtype``. """
+        if K1.gens == K0.symbols:
+            if K1.dom == K0.dom:
+                return K1(dict(a))  # set the correct ring
+            else:
+                convert_dom = lambda c: K1.dom.convert_from(c, K0.dom)
+                return K1._from_dict({m: convert_dom(c) for m, c in a.items()})
+        else:
+            monoms, coeffs = _dict_reorder(a.to_dict(), K0.symbols, K1.gens)
+
+            if K1.dom != K0.dom:
+                coeffs = [ K1.dom.convert(c, K0.dom) for c in coeffs ]
+
+            return K1._from_dict(dict(zip(monoms, coeffs)))
+
+    def from_GlobalPolynomialRing(K1, a, K0):
+        """Convert a ``DMP`` object to ``dtype``. """
+        if K1.gens == K0.gens:
+            if K1.dom != K0.dom:
+                a = a.convert(K1.dom)
+            return K1(a.to_list())
+        else:
+            monoms, coeffs = _dict_reorder(a.to_dict(), K0.gens, K1.gens)
+
+            if K1.dom != K0.dom:
+                coeffs = [ K1.dom.convert(c, K0.dom) for c in coeffs ]
+
+            return K1(dict(zip(monoms, coeffs)))
+
+    def get_field(self):
+        """Returns a field associated with ``self``. """
+        return FractionField(self.dom, *self.gens)
+
+    def poly_ring(self, *gens):
+        """Returns a polynomial ring, i.e. ``K[X]``. """
+        raise NotImplementedError('nested domains not allowed')
+
+    def frac_field(self, *gens):
+        """Returns a fraction field, i.e. ``K(X)``. """
+        raise NotImplementedError('nested domains not allowed')
+
+    def revert(self, a):
+        try:
+            return self.exquo(self.one, a)
+        except (ExactQuotientFailed, ZeroDivisionError):
+            raise NotReversible('%s is not a unit' % a)
+
+    def gcdex(self, a, b):
+        """Extended GCD of ``a`` and ``b``. """
+        return a.gcdex(b)
+
+    def gcd(self, a, b):
+        """Returns GCD of ``a`` and ``b``. """
+        return a.gcd(b)
+
+    def lcm(self, a, b):
+        """Returns LCM of ``a`` and ``b``. """
+        return a.lcm(b)
+
+    def factorial(self, a):
+        """Returns factorial of ``a``. """
+        return self.dtype(self.dom.factorial(a))
+
+    def _vector_to_sdm(self, v, order):
+        """
+        For internal use by the modules class.
+
+        Convert an iterable of elements of this ring into a sparse distributed
+        module element.
+        """
+        raise NotImplementedError
+
+    def _sdm_to_dics(self, s, n):
+        """Helper for _sdm_to_vector."""
+        from sympy.polys.distributedmodules import sdm_to_dict
+        dic = sdm_to_dict(s)
+        res = [{} for _ in range(n)]
+        for k, v in dic.items():
+            res[k[0]][k[1:]] = v
+        return res
+
+    def _sdm_to_vector(self, s, n):
+        """
+        For internal use by the modules class.
+
+        Convert a sparse distributed module into a list of length ``n``.
+
+        Examples
+        ========
+
+        >>> from sympy import QQ, ilex
+        >>> from sympy.abc import x, y
+        >>> R = QQ.old_poly_ring(x, y, order=ilex)
+        >>> L = [((1, 1, 1), QQ(1)), ((0, 1, 0), QQ(1)), ((0, 0, 1), QQ(2))]
+        >>> R._sdm_to_vector(L, 2)
+        [DMF([[1], [2, 0]], [[1]], QQ), DMF([[1, 0], []], [[1]], QQ)]
+        """
+        dics = self._sdm_to_dics(s, n)
+        # NOTE this works for global and local rings!
+        return [self(x) for x in dics]
+
+    def free_module(self, rank):
+        """
+        Generate a free module of rank ``rank`` over ``self``.
+
+        Examples
+        ========
+
+        >>> from sympy.abc import x
+        >>> from sympy import QQ
+        >>> QQ.old_poly_ring(x).free_module(2)
+        QQ[x]**2
+        """
+        return FreeModulePolyRing(self, rank)
+
+
+def _vector_to_sdm_helper(v, order):
+    """Helper method for common code in Global and Local poly rings."""
+    from sympy.polys.distributedmodules import sdm_from_dict
+    d = {}
+    for i, e in enumerate(v):
+        for key, value in e.to_dict().items():
+            d[(i,) + key] = value
+    return sdm_from_dict(d, order)
+
+
+@public
+class GlobalPolynomialRing(PolynomialRingBase):
+    """A true polynomial ring, with objects DMP. """
+
+    is_PolynomialRing = is_Poly = True
+    dtype = DMP
+
+    def new(self, element):
+        if isinstance(element, dict):
+            return DMP.from_dict(element, len(self.gens) - 1, self.dom)
+        elif element in self.dom:
+            return self._ground_new(self.dom.convert(element))
+        else:
+            return self.dtype(element, self.dom, len(self.gens) - 1)
+
+    def from_FractionField(K1, a, K0):
+        """
+        Convert a ``DMF`` object to ``DMP``.
+
+        Examples
+        ========
+
+        >>> from sympy.polys.polyclasses import DMP, DMF
+        >>> from sympy.polys.domains import ZZ
+        >>> from sympy.abc import x
+
+        >>> f = DMF(([ZZ(1), ZZ(1)], [ZZ(1)]), ZZ)
+        >>> K = ZZ.old_frac_field(x)
+
+        >>> F = ZZ.old_poly_ring(x).from_FractionField(f, K)
+
+        >>> F == DMP([ZZ(1), ZZ(1)], ZZ)
+        True
+        >>> type(F)  # doctest: +SKIP
+        <class 'sympy.polys.polyclasses.DMP_Python'>
+
+        """
+        if a.denom().is_one:
+            return K1.from_GlobalPolynomialRing(a.numer(), K0)
+
+    def to_sympy(self, a):
+        """Convert ``a`` to a SymPy object. """
+        return basic_from_dict(a.to_sympy_dict(), *self.gens)
+
+    def from_sympy(self, a):
+        """Convert SymPy's expression to ``dtype``. """
+        try:
+            rep, _ = dict_from_basic(a, gens=self.gens)
+        except PolynomialError:
+            raise CoercionFailed("Cannot convert %s to type %s" % (a, self))
+
+        for k, v in rep.items():
+            rep[k] = self.dom.from_sympy(v)
+
+        return DMP.from_dict(rep, self.ngens - 1, self.dom)
+
+    def is_positive(self, a):
+        """Returns True if ``LC(a)`` is positive. """
+        return self.dom.is_positive(a.LC())
+
+    def is_negative(self, a):
+        """Returns True if ``LC(a)`` is negative. """
+        return self.dom.is_negative(a.LC())
+
+    def is_nonpositive(self, a):
+        """Returns True if ``LC(a)`` is non-positive. """
+        return self.dom.is_nonpositive(a.LC())
+
+    def is_nonnegative(self, a):
+        """Returns True if ``LC(a)`` is non-negative. """
+        return self.dom.is_nonnegative(a.LC())
+
+    def _vector_to_sdm(self, v, order):
+        """
+        Examples
+        ========
+
+        >>> from sympy import lex, QQ
+        >>> from sympy.abc import x, y
+        >>> R = QQ.old_poly_ring(x, y)
+        >>> f = R.convert(x + 2*y)
+        >>> g = R.convert(x * y)
+        >>> R._vector_to_sdm([f, g], lex)
+        [((1, 1, 1), 1), ((0, 1, 0), 1), ((0, 0, 1), 2)]
+        """
+        return _vector_to_sdm_helper(v, order)
+
+
+class GeneralizedPolynomialRing(PolynomialRingBase):
+    """A generalized polynomial ring, with objects DMF. """
+
+    dtype = DMF
+
+    def new(self, a):
+        """Construct an element of ``self`` domain from ``a``. """
+        res = self.dtype(a, self.dom, len(self.gens) - 1)
+
+        # make sure res is actually in our ring
+        if res.denom().terms(order=self.order)[0][0] != (0,)*len(self.gens):
+            from sympy.printing.str import sstr
+            raise CoercionFailed("denominator %s not allowed in %s"
+                                 % (sstr(res), self))
+        return res
+
+    def __contains__(self, a):
+        try:
+            a = self.convert(a)
+        except CoercionFailed:
+            return False
+        return a.denom().terms(order=self.order)[0][0] == (0,)*len(self.gens)
+
+    def to_sympy(self, a):
+        """Convert ``a`` to a SymPy object. """
+        return (basic_from_dict(a.numer().to_sympy_dict(), *self.gens) /
+                basic_from_dict(a.denom().to_sympy_dict(), *self.gens))
+
+    def from_sympy(self, a):
+        """Convert SymPy's expression to ``dtype``. """
+        p, q = a.as_numer_denom()
+
+        num, _ = dict_from_basic(p, gens=self.gens)
+        den, _ = dict_from_basic(q, gens=self.gens)
+
+        for k, v in num.items():
+            num[k] = self.dom.from_sympy(v)
+
+        for k, v in den.items():
+            den[k] = self.dom.from_sympy(v)
+
+        return self((num, den)).cancel()
+
+    def exquo(self, a, b):
+        """Exact quotient of ``a`` and ``b``. """
+        # Elements are DMF that will always divide (except 0). The result is
+        # not guaranteed to be in this ring, so we have to check that.
+        r = a / b
+
+        try:
+            r = self.new((r.num, r.den))
+        except CoercionFailed:
+            raise ExactQuotientFailed(a, b, self)
+
+        return r
+
+    def from_FractionField(K1, a, K0):
+        dmf = K1.get_field().from_FractionField(a, K0)
+        return K1((dmf.num, dmf.den))
+
+    def _vector_to_sdm(self, v, order):
+        """
+        Turn an iterable into a sparse distributed module.
+
+        Note that the vector is multiplied by a unit first to make all entries
+        polynomials.
+
+        Examples
+        ========
+
+        >>> from sympy import ilex, QQ
+        >>> from sympy.abc import x, y
+        >>> R = QQ.old_poly_ring(x, y, order=ilex)
+        >>> f = R.convert((x + 2*y) / (1 + x))
+        >>> g = R.convert(x * y)
+        >>> R._vector_to_sdm([f, g], ilex)
+        [((0, 0, 1), 2), ((0, 1, 0), 1), ((1, 1, 1), 1), ((1,
+          2, 1), 1)]
+        """
+        # NOTE this is quite inefficient...
+        u = self.one.numer()
+        for x in v:
+            u *= x.denom()
+        return _vector_to_sdm_helper([x.numer()*u/x.denom() for x in v], order)
+
+
+@public
+def PolynomialRing(dom, *gens, **opts):
+    r"""
+    Create a generalized multivariate polynomial ring.
+
+    A generalized polynomial ring is defined by a ground field `K`, a set
+    of generators (typically `x_1, \ldots, x_n`) and a monomial order `<`.
+    The monomial order can be global, local or mixed. In any case it induces
+    a total ordering on the monomials, and there exists for every (non-zero)
+    polynomial `f \in K[x_1, \ldots, x_n]` a well-defined "leading monomial"
+    `LM(f) = LM(f, >)`. One can then define a multiplicative subset
+    `S = S_> = \{f \in K[x_1, \ldots, x_n] | LM(f) = 1\}`. The generalized
+    polynomial ring corresponding to the monomial order is
+    `R = S^{-1}K[x_1, \ldots, x_n]`.
+
+    If `>` is a so-called global order, that is `1` is the smallest monomial,
+    then we just have `S = K` and `R = K[x_1, \ldots, x_n]`.
+
+    Examples
+    ========
+
+    A few examples may make this clearer.
+
+    >>> from sympy.abc import x, y
+    >>> from sympy import QQ
+
+    Our first ring uses global lexicographic order.
+
+    >>> R1 = QQ.old_poly_ring(x, y, order=(("lex", x, y),))
+
+    The second ring uses local lexicographic order. Note that when using a
+    single (non-product) order, you can just specify the name and omit the
+    variables:
+
+    >>> R2 = QQ.old_poly_ring(x, y, order="ilex")
+
+    The third and fourth rings use a mixed orders:
+
+    >>> o1 = (("ilex", x), ("lex", y))
+    >>> o2 = (("lex", x), ("ilex", y))
+    >>> R3 = QQ.old_poly_ring(x, y, order=o1)
+    >>> R4 = QQ.old_poly_ring(x, y, order=o2)
+
+    We will investigate what elements of `K(x, y)` are contained in the various
+    rings.
+
+    >>> L = [x, 1/x, y/(1 + x), 1/(1 + y), 1/(1 + x*y)]
+    >>> test = lambda R: [f in R for f in L]
+
+    The first ring is just `K[x, y]`:
+
+    >>> test(R1)
+    [True, False, False, False, False]
+
+    The second ring is R1 localised at the maximal ideal (x, y):
+
+    >>> test(R2)
+    [True, False, True, True, True]
+
+    The third ring is R1 localised at the prime ideal (x):
+
+    >>> test(R3)
+    [True, False, True, False, True]
+
+    Finally the fourth ring is R1 localised at `S = K[x, y] \setminus yK[y]`:
+
+    >>> test(R4)
+    [True, False, False, True, False]
+    """
+
+    order = opts.get("order", GeneralizedPolynomialRing.default_order)
+    if iterable(order):
+        order = build_product_order(order, gens)
+    order = monomial_key(order)
+    opts['order'] = order
+
+    if order.is_global:
+        return GlobalPolynomialRing(dom, *gens, **opts)
+    else:
+        return GeneralizedPolynomialRing(dom, *gens, **opts)

.venv/lib/python3.13/site-packages/sympy/polys/domains/polynomialring.py

@@ -0,0 +1,203 @@
+"""Implementation of :class:`PolynomialRing` class. """
+
+
+from sympy.polys.domains.ring import Ring
+from sympy.polys.domains.compositedomain import CompositeDomain
+
+from sympy.polys.polyerrors import CoercionFailed, GeneratorsError
+from sympy.utilities import public
+
+@public
+class PolynomialRing(Ring, CompositeDomain):
+    """A class for representing multivariate polynomial rings. """
+
+    is_PolynomialRing = is_Poly = True
+
+    has_assoc_Ring  = True
+    has_assoc_Field = True
+
+    def __init__(self, domain_or_ring, symbols=None, order=None):
+        from sympy.polys.rings import PolyRing
+
+        if isinstance(domain_or_ring, PolyRing) and symbols is None and order is None:
+            ring = domain_or_ring
+        else:
+            ring = PolyRing(symbols, domain_or_ring, order)
+
+        self.ring = ring
+        self.dtype = ring.dtype
+
+        self.gens = ring.gens
+        self.ngens = ring.ngens
+        self.symbols = ring.symbols
+        self.domain = ring.domain
+
+
+        if symbols:
+            if ring.domain.is_Field and ring.domain.is_Exact and len(symbols)==1:
+                self.is_PID = True
+
+        # TODO: remove this
+        self.dom = self.domain
+
+    def new(self, element):
+        return self.ring.ring_new(element)
+
+    def of_type(self, element):
+        """Check if ``a`` is of type ``dtype``. """
+        return self.ring.is_element(element)
+
+    @property
+    def zero(self):
+        return self.ring.zero
+
+    @property
+    def one(self):
+        return self.ring.one
+
+    @property
+    def order(self):
+        return self.ring.order
+
+    def __str__(self):
+        return str(self.domain) + '[' + ','.join(map(str, self.symbols)) + ']'
+
+    def __hash__(self):
+        return hash((self.__class__.__name__, self.ring, self.domain, self.symbols))
+
+    def __eq__(self, other):
+        """Returns `True` if two domains are equivalent. """
+        if not isinstance(other, PolynomialRing):
+            return NotImplemented
+        return self.ring == other.ring
+
+    def is_unit(self, a):
+        """Returns ``True`` if ``a`` is a unit of ``self``"""
+        if not a.is_ground:
+            return False
+        K = self.domain
+        return K.is_unit(K.convert_from(a, self))
+
+    def canonical_unit(self, a):
+        u = self.domain.canonical_unit(a.LC)
+        return self.ring.ground_new(u)
+
+    def to_sympy(self, a):
+        """Convert `a` to a SymPy object. """
+        return a.as_expr()
+
+    def from_sympy(self, a):
+        """Convert SymPy's expression to `dtype`. """
+        return self.ring.from_expr(a)
+
+    def from_ZZ(K1, a, K0):
+        """Convert a Python `int` object to `dtype`. """
+        return K1(K1.domain.convert(a, K0))
+
+    def from_ZZ_python(K1, a, K0):
+        """Convert a Python `int` object to `dtype`. """
+        return K1(K1.domain.convert(a, K0))
+
+    def from_QQ(K1, a, K0):
+        """Convert a Python `Fraction` object to `dtype`. """
+        return K1(K1.domain.convert(a, K0))
+
+    def from_QQ_python(K1, a, K0):
+        """Convert a Python `Fraction` object to `dtype`. """
+        return K1(K1.domain.convert(a, K0))
+
+    def from_ZZ_gmpy(K1, a, K0):
+        """Convert a GMPY `mpz` object to `dtype`. """
+        return K1(K1.domain.convert(a, K0))
+
+    def from_QQ_gmpy(K1, a, K0):
+        """Convert a GMPY `mpq` object to `dtype`. """
+        return K1(K1.domain.convert(a, K0))
+
+    def from_GaussianIntegerRing(K1, a, K0):
+        """Convert a `GaussianInteger` object to `dtype`. """
+        return K1(K1.domain.convert(a, K0))
+
+    def from_GaussianRationalField(K1, a, K0):
+        """Convert a `GaussianRational` object to `dtype`. """
+        return K1(K1.domain.convert(a, K0))
+
+    def from_RealField(K1, a, K0):
+        """Convert a mpmath `mpf` object to `dtype`. """
+        return K1(K1.domain.convert(a, K0))
+
+    def from_ComplexField(K1, a, K0):
+        """Convert a mpmath `mpf` object to `dtype`. """
+        return K1(K1.domain.convert(a, K0))
+
+    def from_AlgebraicField(K1, a, K0):
+        """Convert an algebraic number to ``dtype``. """
+        if K1.domain != K0:
+            a = K1.domain.convert_from(a, K0)
+        if a is not None:
+            return K1.new(a)
+
+    def from_PolynomialRing(K1, a, K0):
+        """Convert a polynomial to ``dtype``. """
+        try:
+            return a.set_ring(K1.ring)
+        except (CoercionFailed, GeneratorsError):
+            return None
+
+    def from_FractionField(K1, a, K0):
+        """Convert a rational function to ``dtype``. """
+        if K1.domain == K0:
+            return K1.ring.from_list([a])
+
+        q, r = K0.numer(a).div(K0.denom(a))
+
+        if r.is_zero:
+            return K1.from_PolynomialRing(q, K0.field.ring.to_domain())
+        else:
+            return None
+
+    def from_GlobalPolynomialRing(K1, a, K0):
+        """Convert from old poly ring to ``dtype``. """
+        if K1.symbols == K0.gens:
+            ad = a.to_dict()
+            if K1.domain != K0.domain:
+                ad = {m: K1.domain.convert(c) for m, c in ad.items()}
+            return K1(ad)
+        elif a.is_ground and K0.domain == K1:
+            return K1.convert_from(a.to_list()[0], K0.domain)
+
+    def get_field(self):
+        """Returns a field associated with `self`. """
+        return self.ring.to_field().to_domain()
+
+    def is_positive(self, a):
+        """Returns True if `LC(a)` is positive. """
+        return self.domain.is_positive(a.LC)
+
+    def is_negative(self, a):
+        """Returns True if `LC(a)` is negative. """
+        return self.domain.is_negative(a.LC)
+
+    def is_nonpositive(self, a):
+        """Returns True if `LC(a)` is non-positive. """
+        return self.domain.is_nonpositive(a.LC)
+
+    def is_nonnegative(self, a):
+        """Returns True if `LC(a)` is non-negative. """
+        return self.domain.is_nonnegative(a.LC)
+
+    def gcdex(self, a, b):
+        """Extended GCD of `a` and `b`. """
+        return a.gcdex(b)
+
+    def gcd(self, a, b):
+        """Returns GCD of `a` and `b`. """
+        return a.gcd(b)
+
+    def lcm(self, a, b):
+        """Returns LCM of `a` and `b`. """
+        return a.lcm(b)
+
+    def factorial(self, a):
+        """Returns factorial of `a`. """
+        return self.dtype(self.domain.factorial(a))

.venv/lib/python3.13/site-packages/sympy/polys/domains/pythonfinitefield.py

@@ -0,0 +1,16 @@
+"""Implementation of :class:`PythonFiniteField` class. """
+
+
+from sympy.polys.domains.finitefield import FiniteField
+from sympy.polys.domains.pythonintegerring import PythonIntegerRing
+
+from sympy.utilities import public
+
+@public
+class PythonFiniteField(FiniteField):
+    """Finite field based on Python's integers. """
+
+    alias = 'FF_python'
+
+    def __init__(self, mod, symmetric=True):
+        super().__init__(mod, PythonIntegerRing(), symmetric)

.venv/lib/python3.13/site-packages/sympy/polys/domains/pythonintegerring.py

@@ -0,0 +1,98 @@
+"""Implementation of :class:`PythonIntegerRing` class. """
+
+
+from sympy.core.numbers import int_valued
+from sympy.polys.domains.groundtypes import (
+    PythonInteger, SymPyInteger, sqrt as python_sqrt,
+    factorial as python_factorial, python_gcdex, python_gcd, python_lcm,
+)
+from sympy.polys.domains.integerring import IntegerRing
+from sympy.polys.polyerrors import CoercionFailed
+from sympy.utilities import public
+
+@public
+class PythonIntegerRing(IntegerRing):
+    """Integer ring based on Python's ``int`` type.
+
+    This will be used as :ref:`ZZ` if ``gmpy`` and ``gmpy2`` are not
+    installed. Elements are instances of the standard Python ``int`` type.
+    """
+
+    dtype = PythonInteger
+    zero = dtype(0)
+    one = dtype(1)
+    alias = 'ZZ_python'
+
+    def __init__(self):
+        """Allow instantiation of this domain. """
+
+    def to_sympy(self, a):
+        """Convert ``a`` to a SymPy object. """
+        return SymPyInteger(a)
+
+    def from_sympy(self, a):
+        """Convert SymPy's Integer to ``dtype``. """
+        if a.is_Integer:
+            return PythonInteger(a.p)
+        elif int_valued(a):
+            return PythonInteger(int(a))
+        else:
+            raise CoercionFailed("expected an integer, got %s" % a)
+
+    def from_FF_python(K1, a, K0):
+        """Convert ``ModularInteger(int)`` to Python's ``int``. """
+        return K0.to_int(a)
+
+    def from_ZZ_python(K1, a, K0):
+        """Convert Python's ``int`` to Python's ``int``. """
+        return a
+
+    def from_QQ(K1, a, K0):
+        """Convert Python's ``Fraction`` to Python's ``int``. """
+        if a.denominator == 1:
+            return a.numerator
+
+    def from_QQ_python(K1, a, K0):
+        """Convert Python's ``Fraction`` to Python's ``int``. """
+        if a.denominator == 1:
+            return a.numerator
+
+    def from_FF_gmpy(K1, a, K0):
+        """Convert ``ModularInteger(mpz)`` to Python's ``int``. """
+        return PythonInteger(K0.to_int(a))
+
+    def from_ZZ_gmpy(K1, a, K0):
+        """Convert GMPY's ``mpz`` to Python's ``int``. """
+        return PythonInteger(a)
+
+    def from_QQ_gmpy(K1, a, K0):
+        """Convert GMPY's ``mpq`` to Python's ``int``. """
+        if a.denom() == 1:
+            return PythonInteger(a.numer())
+
+    def from_RealField(K1, a, K0):
+        """Convert mpmath's ``mpf`` to Python's ``int``. """
+        p, q = K0.to_rational(a)
+
+        if q == 1:
+            return PythonInteger(p)
+
+    def gcdex(self, a, b):
+        """Compute extended GCD of ``a`` and ``b``. """
+        return python_gcdex(a, b)
+
+    def gcd(self, a, b):
+        """Compute GCD of ``a`` and ``b``. """
+        return python_gcd(a, b)
+
+    def lcm(self, a, b):
+        """Compute LCM of ``a`` and ``b``. """
+        return python_lcm(a, b)
+
+    def sqrt(self, a):
+        """Compute square root of ``a``. """
+        return python_sqrt(a)
+
+    def factorial(self, a):
+        """Compute factorial of ``a``. """
+        return python_factorial(a)

.venv/lib/python3.13/site-packages/sympy/polys/domains/pythonrational.py

@@ -0,0 +1,22 @@
+"""
+Rational number type based on Python integers.
+
+The PythonRational class from here has been moved to
+sympy.external.pythonmpq
+
+This module is just left here for backwards compatibility.
+"""
+
+
+from sympy.core.numbers import Rational
+from sympy.core.sympify import _sympy_converter
+from sympy.utilities import public
+from sympy.external.pythonmpq import PythonMPQ
+
+
+PythonRational = public(PythonMPQ)
+
+
+def sympify_pythonrational(arg):
+    return Rational(arg.numerator, arg.denominator)
+_sympy_converter[PythonRational] = sympify_pythonrational

.venv/lib/python3.13/site-packages/sympy/polys/domains/pythonrationalfield.py

@@ -0,0 +1,73 @@
+"""Implementation of :class:`PythonRationalField` class. """
+
+
+from sympy.polys.domains.groundtypes import PythonInteger, PythonRational, SymPyRational
+from sympy.polys.domains.rationalfield import RationalField
+from sympy.polys.polyerrors import CoercionFailed
+from sympy.utilities import public
+
+@public
+class PythonRationalField(RationalField):
+    """Rational field based on :ref:`MPQ`.
+
+    This will be used as :ref:`QQ` if ``gmpy`` and ``gmpy2`` are not
+    installed. Elements are instances of :ref:`MPQ`.
+    """
+
+    dtype = PythonRational
+    zero = dtype(0)
+    one = dtype(1)
+    alias = 'QQ_python'
+
+    def __init__(self):
+        pass
+
+    def get_ring(self):
+        """Returns ring associated with ``self``. """
+        from sympy.polys.domains import PythonIntegerRing
+        return PythonIntegerRing()
+
+    def to_sympy(self, a):
+        """Convert `a` to a SymPy object. """
+        return SymPyRational(a.numerator, a.denominator)
+
+    def from_sympy(self, a):
+        """Convert SymPy's Rational to `dtype`. """
+        if a.is_Rational:
+            return PythonRational(a.p, a.q)
+        elif a.is_Float:
+            from sympy.polys.domains import RR
+            p, q = RR.to_rational(a)
+            return PythonRational(int(p), int(q))
+        else:
+            raise CoercionFailed("expected `Rational` object, got %s" % a)
+
+    def from_ZZ_python(K1, a, K0):
+        """Convert a Python `int` object to `dtype`. """
+        return PythonRational(a)
+
+    def from_QQ_python(K1, a, K0):
+        """Convert a Python `Fraction` object to `dtype`. """
+        return a
+
+    def from_ZZ_gmpy(K1, a, K0):
+        """Convert a GMPY `mpz` object to `dtype`. """
+        return PythonRational(PythonInteger(a))
+
+    def from_QQ_gmpy(K1, a, K0):
+        """Convert a GMPY `mpq` object to `dtype`. """
+        return PythonRational(PythonInteger(a.numer()),
+                              PythonInteger(a.denom()))
+
+    def from_RealField(K1, a, K0):
+        """Convert a mpmath `mpf` object to `dtype`. """
+        p, q = K0.to_rational(a)
+        return PythonRational(int(p), int(q))
+
+    def numer(self, a):
+        """Returns numerator of `a`. """
+        return a.numerator
+
+    def denom(self, a):
+        """Returns denominator of `a`. """
+        return a.denominator

.venv/lib/python3.13/site-packages/sympy/polys/domains/quotientring.py

@@ -0,0 +1,202 @@
+"""Implementation of :class:`QuotientRing` class."""
+
+
+from sympy.polys.agca.modules import FreeModuleQuotientRing
+from sympy.polys.domains.ring import Ring
+from sympy.polys.polyerrors import NotReversible, CoercionFailed
+from sympy.utilities import public
+
+# TODO
+# - successive quotients (when quotient ideals are implemented)
+# - poly rings over quotients?
+# - division by non-units in integral domains?
+
+@public
+class QuotientRingElement:
+    """
+    Class representing elements of (commutative) quotient rings.
+
+    Attributes:
+
+    - ring - containing ring
+    - data - element of ring.ring (i.e. base ring) representing self
+    """
+
+    def __init__(self, ring, data):
+        self.ring = ring
+        self.data = data
+
+    def __str__(self):
+        from sympy.printing.str import sstr
+        data = self.ring.ring.to_sympy(self.data)
+        return sstr(data) + " + " + str(self.ring.base_ideal)
+
+    __repr__ = __str__
+
+    def __bool__(self):
+        return not self.ring.is_zero(self)
+
+    def __add__(self, om):
+        if not isinstance(om, self.__class__) or om.ring != self.ring:
+            try:
+                om = self.ring.convert(om)
+            except (NotImplementedError, CoercionFailed):
+                return NotImplemented
+        return self.ring(self.data + om.data)
+
+    __radd__ = __add__
+
+    def __neg__(self):
+        return self.ring(self.data*self.ring.ring.convert(-1))
+
+    def __sub__(self, om):
+        return self.__add__(-om)
+
+    def __rsub__(self, om):
+        return (-self).__add__(om)
+
+    def __mul__(self, o):
+        if not isinstance(o, self.__class__):
+            try:
+                o = self.ring.convert(o)
+            except (NotImplementedError, CoercionFailed):
+                return NotImplemented
+        return self.ring(self.data*o.data)
+
+    __rmul__ = __mul__
+
+    def __rtruediv__(self, o):
+        return self.ring.revert(self)*o
+
+    def __truediv__(self, o):
+        if not isinstance(o, self.__class__):
+            try:
+                o = self.ring.convert(o)
+            except (NotImplementedError, CoercionFailed):
+                return NotImplemented
+        return self.ring.revert(o)*self
+
+    def __pow__(self, oth):
+        if oth < 0:
+            return self.ring.revert(self) ** -oth
+        return self.ring(self.data ** oth)
+
+    def __eq__(self, om):
+        if not isinstance(om, self.__class__) or om.ring != self.ring:
+            return False
+        return self.ring.is_zero(self - om)
+
+    def __ne__(self, om):
+        return not self == om
+
+
+class QuotientRing(Ring):
+    """
+    Class representing (commutative) quotient rings.
+
+    You should not usually instantiate this by hand, instead use the constructor
+    from the base ring in the construction.
+
+    >>> from sympy.abc import x
+    >>> from sympy import QQ
+    >>> I = QQ.old_poly_ring(x).ideal(x**3 + 1)
+    >>> QQ.old_poly_ring(x).quotient_ring(I)
+    QQ[x]/<x**3 + 1>
+
+    Shorter versions are possible:
+
+    >>> QQ.old_poly_ring(x)/I
+    QQ[x]/<x**3 + 1>
+
+    >>> QQ.old_poly_ring(x)/[x**3 + 1]
+    QQ[x]/<x**3 + 1>
+
+    Attributes:
+
+    - ring - the base ring
+    - base_ideal - the ideal used to form the quotient
+    """
+
+    has_assoc_Ring = True
+    has_assoc_Field = False
+    dtype = QuotientRingElement
+
+    def __init__(self, ring, ideal):
+        if not ideal.ring == ring:
+            raise ValueError('Ideal must belong to %s, got %s' % (ring, ideal))
+        self.ring = ring
+        self.base_ideal = ideal
+        self.zero = self(self.ring.zero)
+        self.one = self(self.ring.one)
+
+    def __str__(self):
+        return str(self.ring) + "/" + str(self.base_ideal)
+
+    def __hash__(self):
+        return hash((self.__class__.__name__, self.dtype, self.ring, self.base_ideal))
+
+    def new(self, a):
+        """Construct an element of ``self`` domain from ``a``. """
+        if not isinstance(a, self.ring.dtype):
+            a = self.ring(a)
+        # TODO optionally disable reduction?
+        return self.dtype(self, self.base_ideal.reduce_element(a))
+
+    def __eq__(self, other):
+        """Returns ``True`` if two domains are equivalent. """
+        return isinstance(other, QuotientRing) and \
+            self.ring == other.ring and self.base_ideal == other.base_ideal
+
+    def from_ZZ(K1, a, K0):
+        """Convert a Python ``int`` object to ``dtype``. """
+        return K1(K1.ring.convert(a, K0))
+
+    from_ZZ_python = from_ZZ
+    from_QQ_python = from_ZZ_python
+    from_ZZ_gmpy = from_ZZ_python
+    from_QQ_gmpy = from_ZZ_python
+    from_RealField = from_ZZ_python
+    from_GlobalPolynomialRing = from_ZZ_python
+    from_FractionField = from_ZZ_python
+
+    def from_sympy(self, a):
+        return self(self.ring.from_sympy(a))
+
+    def to_sympy(self, a):
+        return self.ring.to_sympy(a.data)
+
+    def from_QuotientRing(self, a, K0):
+        if K0 == self:
+            return a
+
+    def poly_ring(self, *gens):
+        """Returns a polynomial ring, i.e. ``K[X]``. """
+        raise NotImplementedError('nested domains not allowed')
+
+    def frac_field(self, *gens):
+        """Returns a fraction field, i.e. ``K(X)``. """
+        raise NotImplementedError('nested domains not allowed')
+
+    def revert(self, a):
+        """
+        Compute a**(-1), if possible.
+        """
+        I = self.ring.ideal(a.data) + self.base_ideal
+        try:
+            return self(I.in_terms_of_generators(1)[0])
+        except ValueError:  # 1 not in I
+            raise NotReversible('%s not a unit in %r' % (a, self))
+
+    def is_zero(self, a):
+        return self.base_ideal.contains(a.data)
+
+    def free_module(self, rank):
+        """
+        Generate a free module of rank ``rank`` over ``self``.
+
+        >>> from sympy.abc import x
+        >>> from sympy import QQ
+        >>> (QQ.old_poly_ring(x)/[x**2 + 1]).free_module(2)
+        (QQ[x]/<x**2 + 1>)**2
+        """
+        return FreeModuleQuotientRing(self, rank)

.venv/lib/python3.13/site-packages/sympy/polys/domains/rationalfield.py

@@ -0,0 +1,200 @@
+"""Implementation of :class:`RationalField` class. """
+
+
+from sympy.external.gmpy import MPQ
+
+from sympy.polys.domains.groundtypes import SymPyRational, is_square, sqrtrem
+
+from sympy.polys.domains.characteristiczero import CharacteristicZero
+from sympy.polys.domains.field import Field
+from sympy.polys.domains.simpledomain import SimpleDomain
+from sympy.polys.polyerrors import CoercionFailed
+from sympy.utilities import public
+
+@public
+class RationalField(Field, CharacteristicZero, SimpleDomain):
+    r"""Abstract base class for the domain :ref:`QQ`.
+
+    The :py:class:`RationalField` class represents the field of rational
+    numbers $\mathbb{Q}$ as a :py:class:`~.Domain` in the domain system.
+    :py:class:`RationalField` is a superclass of
+    :py:class:`PythonRationalField` and :py:class:`GMPYRationalField` one of
+    which will be the implementation for :ref:`QQ` depending on whether either
+    of ``gmpy`` or ``gmpy2`` is installed or not.
+
+    See also
+    ========
+
+    Domain
+    """
+
+    rep = 'QQ'
+    alias = 'QQ'
+
+    is_RationalField = is_QQ = True
+    is_Numerical = True
+
+    has_assoc_Ring = True
+    has_assoc_Field = True
+
+    dtype = MPQ
+    zero = dtype(0)
+    one = dtype(1)
+    tp = type(one)
+
+    def __init__(self):
+        pass
+
+    def __eq__(self, other):
+        """Returns ``True`` if two domains are equivalent. """
+        if isinstance(other, RationalField):
+            return True
+        else:
+            return NotImplemented
+
+    def __hash__(self):
+        """Returns hash code of ``self``. """
+        return hash('QQ')
+
+    def get_ring(self):
+        """Returns ring associated with ``self``. """
+        from sympy.polys.domains import ZZ
+        return ZZ
+
+    def to_sympy(self, a):
+        """Convert ``a`` to a SymPy object. """
+        return SymPyRational(int(a.numerator), int(a.denominator))
+
+    def from_sympy(self, a):
+        """Convert SymPy's Integer to ``dtype``. """
+        if a.is_Rational:
+            return MPQ(a.p, a.q)
+        elif a.is_Float:
+            from sympy.polys.domains import RR
+            return MPQ(*map(int, RR.to_rational(a)))
+        else:
+            raise CoercionFailed("expected `Rational` object, got %s" % a)
+
+    def algebraic_field(self, *extension, alias=None):
+        r"""Returns an algebraic field, i.e. `\mathbb{Q}(\alpha, \ldots)`.
+
+        Parameters
+        ==========
+
+        *extension : One or more :py:class:`~.Expr`
+            Generators of the extension. These should be expressions that are
+            algebraic over `\mathbb{Q}`.
+
+        alias : str, :py:class:`~.Symbol`, None, optional (default=None)
+            If provided, this will be used as the alias symbol for the
+            primitive element of the returned :py:class:`~.AlgebraicField`.
+
+        Returns
+        =======
+
+        :py:class:`~.AlgebraicField`
+            A :py:class:`~.Domain` representing the algebraic field extension.
+
+        Examples
+        ========
+
+        >>> from sympy import QQ, sqrt
+        >>> QQ.algebraic_field(sqrt(2))
+        QQ<sqrt(2)>
+        """
+        from sympy.polys.domains import AlgebraicField
+        return AlgebraicField(self, *extension, alias=alias)
+
+    def from_AlgebraicField(K1, a, K0):
+        """Convert a :py:class:`~.ANP` object to :ref:`QQ`.
+
+        See :py:meth:`~.Domain.convert`
+        """
+        if a.is_ground:
+            return K1.convert(a.LC(), K0.dom)
+
+    def from_ZZ(K1, a, K0):
+        """Convert a Python ``int`` object to ``dtype``. """
+        return MPQ(a)
+
+    def from_ZZ_python(K1, a, K0):
+        """Convert a Python ``int`` object to ``dtype``. """
+        return MPQ(a)
+
+    def from_QQ(K1, a, K0):
+        """Convert a Python ``Fraction`` object to ``dtype``. """
+        return MPQ(a.numerator, a.denominator)
+
+    def from_QQ_python(K1, a, K0):
+        """Convert a Python ``Fraction`` object to ``dtype``. """
+        return MPQ(a.numerator, a.denominator)
+
+    def from_ZZ_gmpy(K1, a, K0):
+        """Convert a GMPY ``mpz`` object to ``dtype``. """
+        return MPQ(a)
+
+    def from_QQ_gmpy(K1, a, K0):
+        """Convert a GMPY ``mpq`` object to ``dtype``. """
+        return a
+
+    def from_GaussianRationalField(K1, a, K0):
+        """Convert a ``GaussianElement`` object to ``dtype``. """
+        if a.y == 0:
+            return MPQ(a.x)
+
+    def from_RealField(K1, a, K0):
+        """Convert a mpmath ``mpf`` object to ``dtype``. """
+        return MPQ(*map(int, K0.to_rational(a)))
+
+    def exquo(self, a, b):
+        """Exact quotient of ``a`` and ``b``, implies ``__truediv__``.  """
+        return MPQ(a) / MPQ(b)
+
+    def quo(self, a, b):
+        """Quotient of ``a`` and ``b``, implies ``__truediv__``. """
+        return MPQ(a) / MPQ(b)
+
+    def rem(self, a, b):
+        """Remainder of ``a`` and ``b``, implies nothing.  """
+        return self.zero
+
+    def div(self, a, b):
+        """Division of ``a`` and ``b``, implies ``__truediv__``. """
+        return MPQ(a) / MPQ(b), self.zero
+
+    def numer(self, a):
+        """Returns numerator of ``a``. """
+        return a.numerator
+
+    def denom(self, a):
+        """Returns denominator of ``a``. """
+        return a.denominator
+
+    def is_square(self, a):
+        """Return ``True`` if ``a`` is a square.
+
+        Explanation
+        ===========
+        A rational number is a square if and only if there exists
+        a rational number ``b`` such that ``b * b == a``.
+        """
+        return is_square(a.numerator) and is_square(a.denominator)
+
+    def exsqrt(self, a):
+        """Non-negative square root of ``a`` if ``a`` is a square.
+
+        See also
+        ========
+        is_square
+        """
+        if a.numerator < 0:  # denominator is always positive
+            return None
+        p_sqrt, p_rem = sqrtrem(a.numerator)
+        if p_rem != 0:
+            return None
+        q_sqrt, q_rem = sqrtrem(a.denominator)
+        if q_rem != 0:
+            return None
+        return MPQ(p_sqrt, q_sqrt)
+
+QQ = RationalField()

.venv/lib/python3.13/site-packages/sympy/polys/domains/realfield.py

@@ -0,0 +1,220 @@
+"""Implementation of :class:`RealField` class. """
+
+
+from sympy.external.gmpy import SYMPY_INTS, MPQ
+from sympy.core.numbers import Float
+from sympy.polys.domains.field import Field
+from sympy.polys.domains.simpledomain import SimpleDomain
+from sympy.polys.domains.characteristiczero import CharacteristicZero
+from sympy.polys.polyerrors import CoercionFailed
+from sympy.utilities import public
+
+from mpmath import MPContext
+from mpmath.libmp import to_rational as _mpmath_to_rational
+
+
+def to_rational(s, max_denom, limit=True):
+
+    p, q = _mpmath_to_rational(s._mpf_)
+
+    # Needed for GROUND_TYPES=flint if gmpy2 is installed because mpmath's
+    # to_rational() function returns a gmpy2.mpz instance and if MPQ is
+    # flint.fmpq then MPQ(p, q) will fail.
+    p = int(p)
+    q = int(q)
+
+    if not limit or q <= max_denom:
+        return p, q
+
+    p0, q0, p1, q1 = 0, 1, 1, 0
+    n, d = p, q
+
+    while True:
+        a = n//d
+        q2 = q0 + a*q1
+        if q2 > max_denom:
+            break
+        p0, q0, p1, q1 = p1, q1, p0 + a*p1, q2
+        n, d = d, n - a*d
+
+    k = (max_denom - q0)//q1
+
+    number = MPQ(p, q)
+    bound1 = MPQ(p0 + k*p1, q0 + k*q1)
+    bound2 = MPQ(p1, q1)
+
+    if not bound2 or not bound1:
+        return p, q
+    elif abs(bound2 - number) <= abs(bound1 - number):
+        return bound2.numerator, bound2.denominator
+    else:
+        return bound1.numerator, bound1.denominator
+
+
+@public
+class RealField(Field, CharacteristicZero, SimpleDomain):
+    """Real numbers up to the given precision. """
+
+    rep = 'RR'
+
+    is_RealField = is_RR = True
+
+    is_Exact = False
+    is_Numerical = True
+    is_PID = False
+
+    has_assoc_Ring = False
+    has_assoc_Field = True
+
+    _default_precision = 53
+
+    @property
+    def has_default_precision(self):
+        return self.precision == self._default_precision
+
+    @property
+    def precision(self):
+        return self._context.prec
+
+    @property
+    def dps(self):
+        return self._context.dps
+
+    @property
+    def tolerance(self):
+        return self._tolerance
+
+    def __init__(self, prec=None, dps=None, tol=None):
+        # XXX: The tol parameter is ignored but is kept for now for backwards
+        # compatibility.
+
+        context = MPContext()
+
+        if prec is None and dps is None:
+            context.prec = self._default_precision
+        elif dps is None:
+            context.prec = prec
+        elif prec is None:
+            context.dps = dps
+        else:
+            raise TypeError("Cannot set both prec and dps")
+
+        self._context = context
+
+        self._dtype = context.mpf
+        self.zero = self.dtype(0)
+        self.one = self.dtype(1)
+
+        # Only max_denom here is used for anything and is only used for
+        # to_rational.
+        self._max_denom = max(2**context.prec // 200, 99)
+        self._tolerance = self.one / self._max_denom
+
+    @property
+    def tp(self):
+        # XXX: Domain treats tp as an alias of dtype. Here we need to two
+        # separate things: dtype is a callable to make/convert instances.
+        # We use tp with isinstance to check if an object is an instance
+        # of the domain already.
+        return self._dtype
+
+    def dtype(self, arg):
+        # XXX: This is needed because mpmath does not recognise fmpz.
+        # It might be better to add conversion routines to mpmath and if that
+        # happens then this can be removed.
+        if isinstance(arg, SYMPY_INTS):
+            arg = int(arg)
+        return self._dtype(arg)
+
+    def __eq__(self, other):
+        return isinstance(other, RealField) and self.precision == other.precision
+
+    def __hash__(self):
+        return hash((self.__class__.__name__, self._dtype, self.precision))
+
+    def to_sympy(self, element):
+        """Convert ``element`` to SymPy number. """
+        return Float(element, self.dps)
+
+    def from_sympy(self, expr):
+        """Convert SymPy's number to ``dtype``. """
+        number = expr.evalf(n=self.dps)
+
+        if number.is_Number:
+            return self.dtype(number)
+        else:
+            raise CoercionFailed("expected real number, got %s" % expr)
+
+    def from_ZZ(self, element, base):
+        return self.dtype(element)
+
+    def from_ZZ_python(self, element, base):
+        return self.dtype(element)
+
+    def from_ZZ_gmpy(self, element, base):
+        return self.dtype(int(element))
+
+    # XXX: We need to convert the denominators to int here because mpmath does
+    # not recognise mpz. Ideally mpmath would handle this and if it changed to
+    # do so then the calls to int here could be removed.
+
+    def from_QQ(self, element, base):
+        return self.dtype(element.numerator) / int(element.denominator)
+
+    def from_QQ_python(self, element, base):
+        return self.dtype(element.numerator) / int(element.denominator)
+
+    def from_QQ_gmpy(self, element, base):
+        return self.dtype(int(element.numerator)) / int(element.denominator)
+
+    def from_AlgebraicField(self, element, base):
+        return self.from_sympy(base.to_sympy(element).evalf(self.dps))
+
+    def from_RealField(self, element, base):
+        return self.dtype(element)
+
+    def from_ComplexField(self, element, base):
+        if not element.imag:
+            return self.dtype(element.real)
+
+    def to_rational(self, element, limit=True):
+        """Convert a real number to rational number. """
+        return to_rational(element, self._max_denom, limit=limit)
+
+    def get_ring(self):
+        """Returns a ring associated with ``self``. """
+        return self
+
+    def get_exact(self):
+        """Returns an exact domain associated with ``self``. """
+        from sympy.polys.domains import QQ
+        return QQ
+
+    def gcd(self, a, b):
+        """Returns GCD of ``a`` and ``b``. """
+        return self.one
+
+    def lcm(self, a, b):
+        """Returns LCM of ``a`` and ``b``. """
+        return a*b
+
+    def almosteq(self, a, b, tolerance=None):
+        """Check if ``a`` and ``b`` are almost equal. """
+        return self._context.almosteq(a, b, tolerance)
+
+    def is_square(self, a):
+        """Returns ``True`` if ``a >= 0`` and ``False`` otherwise. """
+        return a >= 0
+
+    def exsqrt(self, a):
+        """Non-negative square root for ``a >= 0`` and ``None`` otherwise.
+
+        Explanation
+        ===========
+        The square root may be slightly inaccurate due to floating point
+        rounding error.
+        """
+        return a ** 0.5 if a >= 0 else None
+
+
+RR = RealField()

.venv/lib/python3.13/site-packages/sympy/polys/domains/ring.py

@@ -0,0 +1,118 @@
+"""Implementation of :class:`Ring` class. """
+
+
+from sympy.polys.domains.domain import Domain
+from sympy.polys.polyerrors import ExactQuotientFailed, NotInvertible, NotReversible
+
+from sympy.utilities import public
+
+@public
+class Ring(Domain):
+    """Represents a ring domain. """
+
+    is_Ring = True
+
+    def get_ring(self):
+        """Returns a ring associated with ``self``. """
+        return self
+
+    def exquo(self, a, b):
+        """Exact quotient of ``a`` and ``b``, implies ``__floordiv__``.  """
+        if a % b:
+            raise ExactQuotientFailed(a, b, self)
+        else:
+            return a // b
+
+    def quo(self, a, b):
+        """Quotient of ``a`` and ``b``, implies ``__floordiv__``. """
+        return a // b
+
+    def rem(self, a, b):
+        """Remainder of ``a`` and ``b``, implies ``__mod__``.  """
+        return a % b
+
+    def div(self, a, b):
+        """Division of ``a`` and ``b``, implies ``__divmod__``. """
+        return divmod(a, b)
+
+    def invert(self, a, b):
+        """Returns inversion of ``a mod b``. """
+        s, t, h = self.gcdex(a, b)
+
+        if self.is_one(h):
+            return s % b
+        else:
+            raise NotInvertible("zero divisor")
+
+    def revert(self, a):
+        """Returns ``a**(-1)`` if possible. """
+        if self.is_one(a) or self.is_one(-a):
+            return a
+        else:
+            raise NotReversible('only units are reversible in a ring')
+
+    def is_unit(self, a):
+        try:
+            self.revert(a)
+            return True
+        except NotReversible:
+            return False
+
+    def numer(self, a):
+        """Returns numerator of ``a``. """
+        return a
+
+    def denom(self, a):
+        """Returns denominator of `a`. """
+        return self.one
+
+    def free_module(self, rank):
+        """
+        Generate a free module of rank ``rank`` over self.
+
+        >>> from sympy.abc import x
+        >>> from sympy import QQ
+        >>> QQ.old_poly_ring(x).free_module(2)
+        QQ[x]**2
+        """
+        raise NotImplementedError
+
+    def ideal(self, *gens):
+        """
+        Generate an ideal of ``self``.
+
+        >>> from sympy.abc import x
+        >>> from sympy import QQ
+        >>> QQ.old_poly_ring(x).ideal(x**2)
+        <x**2>
+        """
+        from sympy.polys.agca.ideals import ModuleImplementedIdeal
+        return ModuleImplementedIdeal(self, self.free_module(1).submodule(
+            *[[x] for x in gens]))
+
+    def quotient_ring(self, e):
+        """
+        Form a quotient ring of ``self``.
+
+        Here ``e`` can be an ideal or an iterable.
+
+        >>> from sympy.abc import x
+        >>> from sympy import QQ
+        >>> QQ.old_poly_ring(x).quotient_ring(QQ.old_poly_ring(x).ideal(x**2))
+        QQ[x]/<x**2>
+        >>> QQ.old_poly_ring(x).quotient_ring([x**2])
+        QQ[x]/<x**2>
+
+        The division operator has been overloaded for this:
+
+        >>> QQ.old_poly_ring(x)/[x**2]
+        QQ[x]/<x**2>
+        """
+        from sympy.polys.agca.ideals import Ideal
+        from sympy.polys.domains.quotientring import QuotientRing
+        if not isinstance(e, Ideal):
+            e = self.ideal(*e)
+        return QuotientRing(self, e)
+
+    def __truediv__(self, e):
+        return self.quotient_ring(e)

.venv/lib/python3.13/site-packages/sympy/polys/domains/simpledomain.py

@@ -0,0 +1,15 @@
+"""Implementation of :class:`SimpleDomain` class. """
+
+
+from sympy.polys.domains.domain import Domain
+from sympy.utilities import public
+
+@public
+class SimpleDomain(Domain):
+    """Base class for simple domains, e.g. ZZ, QQ. """
+
+    is_Simple = True
+
+    def inject(self, *gens):
+        """Inject generators into this domain. """
+        return self.poly_ring(*gens)

.venv/lib/python3.13/site-packages/sympy/polys/tests/test_appellseqs.py

@@ -0,0 +1,91 @@
+"""Tests for efficient functions for generating Appell sequences."""
+from sympy.core.numbers import Rational as Q
+from sympy.polys.polytools import Poly
+from sympy.testing.pytest import raises
+from sympy.polys.appellseqs import (bernoulli_poly, bernoulli_c_poly,
+    euler_poly, genocchi_poly, andre_poly)
+from sympy.abc import x
+
+def test_bernoulli_poly():
+    raises(ValueError, lambda: bernoulli_poly(-1, x))
+    assert bernoulli_poly(1, x, polys=True) == Poly(x - Q(1,2))
+
+    assert bernoulli_poly(0, x) == 1
+    assert bernoulli_poly(1, x) == x - Q(1,2)
+    assert bernoulli_poly(2, x) == x**2 - x + Q(1,6)
+    assert bernoulli_poly(3, x) == x**3 - Q(3,2)*x**2 + Q(1,2)*x
+    assert bernoulli_poly(4, x) == x**4 - 2*x**3 + x**2 - Q(1,30)
+    assert bernoulli_poly(5, x) == x**5 - Q(5,2)*x**4 + Q(5,3)*x**3 - Q(1,6)*x
+    assert bernoulli_poly(6, x) == x**6 - 3*x**5 + Q(5,2)*x**4 - Q(1,2)*x**2 + Q(1,42)
+
+    assert bernoulli_poly(1).dummy_eq(x - Q(1,2))
+    assert bernoulli_poly(1, polys=True) == Poly(x - Q(1,2))
+
+def test_bernoulli_c_poly():
+    raises(ValueError, lambda: bernoulli_c_poly(-1, x))
+    assert bernoulli_c_poly(1, x, polys=True) == Poly(x, domain='QQ')
+
+    assert bernoulli_c_poly(0, x) == 1
+    assert bernoulli_c_poly(1, x) == x
+    assert bernoulli_c_poly(2, x) == x**2 - Q(1,3)
+    assert bernoulli_c_poly(3, x) == x**3 - x
+    assert bernoulli_c_poly(4, x) == x**4 - 2*x**2 + Q(7,15)
+    assert bernoulli_c_poly(5, x) == x**5 - Q(10,3)*x**3 + Q(7,3)*x
+    assert bernoulli_c_poly(6, x) == x**6 - 5*x**4 + 7*x**2 - Q(31,21)
+
+    assert bernoulli_c_poly(1).dummy_eq(x)
+    assert bernoulli_c_poly(1, polys=True) == Poly(x, domain='QQ')
+
+    assert 2**8 * bernoulli_poly(8, (x+1)/2).expand() == bernoulli_c_poly(8, x)
+    assert 2**9 * bernoulli_poly(9, (x+1)/2).expand() == bernoulli_c_poly(9, x)
+
+def test_genocchi_poly():
+    raises(ValueError, lambda: genocchi_poly(-1, x))
+    assert genocchi_poly(2, x, polys=True) == Poly(-2*x + 1)
+
+    assert genocchi_poly(0, x) == 0
+    assert genocchi_poly(1, x) == -1
+    assert genocchi_poly(2, x) == 1 - 2*x
+    assert genocchi_poly(3, x) == 3*x - 3*x**2
+    assert genocchi_poly(4, x) == -1 + 6*x**2 - 4*x**3
+    assert genocchi_poly(5, x) == -5*x + 10*x**3 - 5*x**4
+    assert genocchi_poly(6, x) == 3 - 15*x**2 + 15*x**4 - 6*x**5
+
+    assert genocchi_poly(2).dummy_eq(-2*x + 1)
+    assert genocchi_poly(2, polys=True) == Poly(-2*x + 1)
+
+    assert 2 * (bernoulli_poly(8, x) - bernoulli_c_poly(8, x)) == genocchi_poly(8, x)
+    assert 2 * (bernoulli_poly(9, x) - bernoulli_c_poly(9, x)) == genocchi_poly(9, x)
+
+def test_euler_poly():
+    raises(ValueError, lambda: euler_poly(-1, x))
+    assert euler_poly(1, x, polys=True) == Poly(x - Q(1,2))
+
+    assert euler_poly(0, x) == 1
+    assert euler_poly(1, x) == x - Q(1,2)
+    assert euler_poly(2, x) == x**2 - x
+    assert euler_poly(3, x) == x**3 - Q(3,2)*x**2 + Q(1,4)
+    assert euler_poly(4, x) == x**4 - 2*x**3 + x
+    assert euler_poly(5, x) == x**5 - Q(5,2)*x**4 + Q(5,2)*x**2 - Q(1,2)
+    assert euler_poly(6, x) == x**6 - 3*x**5 + 5*x**3 - 3*x
+
+    assert euler_poly(1).dummy_eq(x - Q(1,2))
+    assert euler_poly(1, polys=True) == Poly(x - Q(1,2))
+
+    assert genocchi_poly(9, x) == euler_poly(8, x) * -9
+    assert genocchi_poly(10, x) == euler_poly(9, x) * -10
+
+def test_andre_poly():
+    raises(ValueError, lambda: andre_poly(-1, x))
+    assert andre_poly(1, x, polys=True) == Poly(x)
+
+    assert andre_poly(0, x) == 1
+    assert andre_poly(1, x) == x
+    assert andre_poly(2, x) == x**2 - 1
+    assert andre_poly(3, x) == x**3 - 3*x
+    assert andre_poly(4, x) == x**4 - 6*x**2 + 5
+    assert andre_poly(5, x) == x**5 - 10*x**3 + 25*x
+    assert andre_poly(6, x) == x**6 - 15*x**4 + 75*x**2 - 61
+
+    assert andre_poly(1).dummy_eq(x)
+    assert andre_poly(1, polys=True) == Poly(x)

.venv/lib/python3.13/site-packages/sympy/polys/tests/test_constructor.py

@@ -0,0 +1,236 @@
+"""Tests for tools for constructing domains for expressions. """
+
+from sympy.testing.pytest import tooslow
+
+from sympy.polys.constructor import construct_domain
+from sympy.polys.domains import ZZ, QQ, ZZ_I, QQ_I, RR, CC, EX
+from sympy.polys.domains.realfield import RealField
+from sympy.polys.domains.complexfield import ComplexField
+
+from sympy.core import (Catalan, GoldenRatio)
+from sympy.core.numbers import (E, Float, I, Rational, pi)
+from sympy.core.singleton import S
+from sympy.functions.elementary.exponential import exp
+from sympy.functions.elementary.miscellaneous import sqrt
+from sympy.functions.elementary.trigonometric import sin
+from sympy import rootof
+
+from sympy.abc import x, y
+
+
+def test_construct_domain():
+
+    assert construct_domain([1, 2, 3]) == (ZZ, [ZZ(1), ZZ(2), ZZ(3)])
+    assert construct_domain([1, 2, 3], field=True) == (QQ, [QQ(1), QQ(2), QQ(3)])
+
+    assert construct_domain([S.One, S(2), S(3)]) == (ZZ, [ZZ(1), ZZ(2), ZZ(3)])
+    assert construct_domain([S.One, S(2), S(3)], field=True) == (QQ, [QQ(1), QQ(2), QQ(3)])
+
+    assert construct_domain([S.Half, S(2)]) == (QQ, [QQ(1, 2), QQ(2)])
+    result = construct_domain([3.14, 1, S.Half])
+    assert isinstance(result[0], RealField)
+    assert result[1] == [RR(3.14), RR(1.0), RR(0.5)]
+
+    result = construct_domain([3.14, I, S.Half])
+    assert isinstance(result[0], ComplexField)
+    assert result[1] == [CC(3.14), CC(1.0j), CC(0.5)]
+
+    assert construct_domain([1.0+I]) == (CC, [CC(1.0, 1.0)])
+    assert construct_domain([2.0+3.0*I]) == (CC, [CC(2.0, 3.0)])
+
+    assert construct_domain([1, I]) == (ZZ_I, [ZZ_I(1, 0), ZZ_I(0, 1)])
+    assert construct_domain([1, I/2]) == (QQ_I, [QQ_I(1, 0), QQ_I(0, S.Half)])
+
+    assert construct_domain([3.14, sqrt(2)], extension=None) == (EX, [EX(3.14), EX(sqrt(2))])
+    assert construct_domain([3.14, sqrt(2)], extension=True) == (EX, [EX(3.14), EX(sqrt(2))])
+
+    assert construct_domain([1, sqrt(2)], extension=None) == (EX, [EX(1), EX(sqrt(2))])
+
+    assert construct_domain([x, sqrt(x)]) == (EX, [EX(x), EX(sqrt(x))])
+    assert construct_domain([x, sqrt(x), sqrt(y)]) == (EX, [EX(x), EX(sqrt(x)), EX(sqrt(y))])
+
+    alg = QQ.algebraic_field(sqrt(2))
+
+    assert construct_domain([7, S.Half, sqrt(2)], extension=True) == \
+        (alg, [alg.convert(7), alg.convert(S.Half), alg.convert(sqrt(2))])
+
+    alg = QQ.algebraic_field(sqrt(2) + sqrt(3))
+
+    assert construct_domain([7, sqrt(2), sqrt(3)], extension=True) == \
+        (alg, [alg.convert(7), alg.convert(sqrt(2)), alg.convert(sqrt(3))])
+
+    dom = ZZ[x]
+
+    assert construct_domain([2*x, 3]) == \
+        (dom, [dom.convert(2*x), dom.convert(3)])
+
+    dom = ZZ[x, y]
+
+    assert construct_domain([2*x, 3*y]) == \
+        (dom, [dom.convert(2*x), dom.convert(3*y)])
+
+    dom = QQ[x]
+
+    assert construct_domain([x/2, 3]) == \
+        (dom, [dom.convert(x/2), dom.convert(3)])
+
+    dom = QQ[x, y]
+
+    assert construct_domain([x/2, 3*y]) == \
+        (dom, [dom.convert(x/2), dom.convert(3*y)])
+
+    dom = ZZ_I[x]
+
+    assert construct_domain([2*x, I]) == \
+        (dom, [dom.convert(2*x), dom.convert(I)])
+
+    dom = ZZ_I[x, y]
+
+    assert construct_domain([2*x, I*y]) == \
+        (dom, [dom.convert(2*x), dom.convert(I*y)])
+
+    dom = QQ_I[x]
+
+    assert construct_domain([x/2, I]) == \
+        (dom, [dom.convert(x/2), dom.convert(I)])
+
+    dom = QQ_I[x, y]
+
+    assert construct_domain([x/2, I*y]) == \
+        (dom, [dom.convert(x/2), dom.convert(I*y)])
+
+    dom = RR[x]
+
+    assert construct_domain([x/2, 3.5]) == \
+        (dom, [dom.convert(x/2), dom.convert(3.5)])
+
+    dom = RR[x, y]
+
+    assert construct_domain([x/2, 3.5*y]) == \
+        (dom, [dom.convert(x/2), dom.convert(3.5*y)])
+
+    dom = CC[x]
+
+    assert construct_domain([I*x/2, 3.5]) == \
+        (dom, [dom.convert(I*x/2), dom.convert(3.5)])
+
+    dom = CC[x, y]
+
+    assert construct_domain([I*x/2, 3.5*y]) == \
+        (dom, [dom.convert(I*x/2), dom.convert(3.5*y)])
+
+    dom = CC[x]
+
+    assert construct_domain([x/2, I*3.5]) == \
+        (dom, [dom.convert(x/2), dom.convert(I*3.5)])
+
+    dom = CC[x, y]
+
+    assert construct_domain([x/2, I*3.5*y]) == \
+        (dom, [dom.convert(x/2), dom.convert(I*3.5*y)])
+
+    dom = ZZ.frac_field(x)
+
+    assert construct_domain([2/x, 3]) == \
+        (dom, [dom.convert(2/x), dom.convert(3)])
+
+    dom = ZZ.frac_field(x, y)
+
+    assert construct_domain([2/x, 3*y]) == \
+        (dom, [dom.convert(2/x), dom.convert(3*y)])
+
+    dom = RR.frac_field(x)
+
+    assert construct_domain([2/x, 3.5]) == \
+        (dom, [dom.convert(2/x), dom.convert(3.5)])
+
+    dom = RR.frac_field(x, y)
+
+    assert construct_domain([2/x, 3.5*y]) == \
+        (dom, [dom.convert(2/x), dom.convert(3.5*y)])
+
+    dom = RealField(prec=336)[x]
+
+    assert construct_domain([pi.evalf(100)*x]) == \
+        (dom, [dom.convert(pi.evalf(100)*x)])
+
+    assert construct_domain(2) == (ZZ, ZZ(2))
+    assert construct_domain(S(2)/3) == (QQ, QQ(2, 3))
+    assert construct_domain(Rational(2, 3)) == (QQ, QQ(2, 3))
+
+    assert construct_domain({}) == (ZZ, {})
+
+
+def test_complex_exponential():
+    w = exp(-I*2*pi/3, evaluate=False)
+    alg = QQ.algebraic_field(w)
+    assert construct_domain([w**2, w, 1], extension=True) == (
+        alg,
+        [alg.convert(w**2),
+         alg.convert(w),
+         alg.convert(1)]
+    )
+
+
+def test_rootof():
+    r1 = rootof(x**3 + x + 1, 0)
+    r2 = rootof(x**3 + x + 1, 1)
+    K1 = QQ.algebraic_field(r1)
+    K2 = QQ.algebraic_field(r2)
+    assert construct_domain([r1]) == (EX, [EX(r1)])
+    assert construct_domain([r2]) == (EX, [EX(r2)])
+    assert construct_domain([r1, r2]) == (EX, [EX(r1), EX(r2)])
+
+    assert construct_domain([r1], extension=True) == (
+            K1, [K1.from_sympy(r1)])
+    assert construct_domain([r2], extension=True) == (
+            K2, [K2.from_sympy(r2)])
+
+
+@tooslow
+def test_rootof_primitive_element():
+    r1 = rootof(x**3 + x + 1, 0)
+    r2 = rootof(x**3 + x + 1, 1)
+    K12 = QQ.algebraic_field(r1 + r2)
+    assert construct_domain([r1, r2], extension=True) == (
+            K12, [K12.from_sympy(r1), K12.from_sympy(r2)])
+
+
+def test_composite_option():
+    assert construct_domain({(1,): sin(y)}, composite=False) == \
+        (EX, {(1,): EX(sin(y))})
+
+    assert construct_domain({(1,): y}, composite=False) == \
+        (EX, {(1,): EX(y)})
+
+    assert construct_domain({(1, 1): 1}, composite=False) == \
+        (ZZ, {(1, 1): 1})
+
+    assert construct_domain({(1, 0): y}, composite=False) == \
+        (EX, {(1, 0): EX(y)})
+
+
+def test_precision():
+    f1 = Float("1.01")
+    f2 = Float("1.0000000000000000000001")
+    for u in [1, 1e-2, 1e-6, 1e-13, 1e-14, 1e-16, 1e-20, 1e-100, 1e-300,
+            f1, f2]:
+        result = construct_domain([u])
+        v = float(result[1][0])
+        assert abs(u - v) / u < 1e-14  # Test relative accuracy
+
+    result = construct_domain([f1])
+    y = result[1][0]
+    assert y-1 > 1e-50
+
+    result = construct_domain([f2])
+    y = result[1][0]
+    assert y-1 > 1e-50
+
+
+def test_issue_11538():
+    for n in [E, pi, Catalan]:
+        assert construct_domain(n)[0] == ZZ[n]
+        assert construct_domain(x + n)[0] == ZZ[x, n]
+    assert construct_domain(GoldenRatio)[0] == EX
+    assert construct_domain(x + GoldenRatio)[0] == EX

.venv/lib/python3.13/site-packages/sympy/polys/tests/test_densearith.py

@@ -0,0 +1,1007 @@
+"""Tests for dense recursive polynomials' arithmetics. """
+
+from sympy.external.gmpy import GROUND_TYPES
+
+from sympy.polys.densebasic import (
+    dup_normal, dmp_normal,
+)
+
+from sympy.polys.densearith import (
+    dup_add_term, dmp_add_term,
+    dup_sub_term, dmp_sub_term,
+    dup_mul_term, dmp_mul_term,
+    dup_add_ground, dmp_add_ground,
+    dup_sub_ground, dmp_sub_ground,
+    dup_mul_ground, dmp_mul_ground,
+    dup_quo_ground, dmp_quo_ground,
+    dup_exquo_ground, dmp_exquo_ground,
+    dup_lshift, dup_rshift,
+    dup_abs, dmp_abs,
+    dup_neg, dmp_neg,
+    dup_add, dmp_add,
+    dup_sub, dmp_sub,
+    dup_mul, dmp_mul,
+    dup_sqr, dmp_sqr,
+    dup_pow, dmp_pow,
+    dup_add_mul, dmp_add_mul,
+    dup_sub_mul, dmp_sub_mul,
+    dup_pdiv, dup_prem, dup_pquo, dup_pexquo,
+    dmp_pdiv, dmp_prem, dmp_pquo, dmp_pexquo,
+    dup_rr_div, dmp_rr_div,
+    dup_ff_div, dmp_ff_div,
+    dup_div, dup_rem, dup_quo, dup_exquo,
+    dmp_div, dmp_rem, dmp_quo, dmp_exquo,
+    dup_max_norm, dmp_max_norm,
+    dup_l1_norm, dmp_l1_norm,
+    dup_l2_norm_squared, dmp_l2_norm_squared,
+    dup_expand, dmp_expand,
+)
+
+from sympy.polys.polyerrors import (
+    ExactQuotientFailed,
+)
+
+from sympy.polys.specialpolys import f_polys, Symbol, Poly
+from sympy.polys.domains import FF, ZZ, QQ, CC
+
+from sympy.testing.pytest import raises
+
+x = Symbol('x')
+
+f_0, f_1, f_2, f_3, f_4, f_5, f_6 = [ f.to_dense() for f in f_polys() ]
+F_0 = dmp_mul_ground(dmp_normal(f_0, 2, QQ), QQ(1, 7), 2, QQ)
+
+def test_dup_add_term():
+    f = dup_normal([], ZZ)
+
+    assert dup_add_term(f, ZZ(0), 0, ZZ) == dup_normal([], ZZ)
+
+    assert dup_add_term(f, ZZ(1), 0, ZZ) == dup_normal([1], ZZ)
+    assert dup_add_term(f, ZZ(1), 1, ZZ) == dup_normal([1, 0], ZZ)
+    assert dup_add_term(f, ZZ(1), 2, ZZ) == dup_normal([1, 0, 0], ZZ)
+
+    f = dup_normal([1, 1, 1], ZZ)
+
+    assert dup_add_term(f, ZZ(1), 0, ZZ) == dup_normal([1, 1, 2], ZZ)
+    assert dup_add_term(f, ZZ(1), 1, ZZ) == dup_normal([1, 2, 1], ZZ)
+    assert dup_add_term(f, ZZ(1), 2, ZZ) == dup_normal([2, 1, 1], ZZ)
+
+    assert dup_add_term(f, ZZ(1), 3, ZZ) == dup_normal([1, 1, 1, 1], ZZ)
+    assert dup_add_term(f, ZZ(1), 4, ZZ) == dup_normal([1, 0, 1, 1, 1], ZZ)
+    assert dup_add_term(f, ZZ(1), 5, ZZ) == dup_normal([1, 0, 0, 1, 1, 1], ZZ)
+    assert dup_add_term(
+        f, ZZ(1), 6, ZZ) == dup_normal([1, 0, 0, 0, 1, 1, 1], ZZ)
+
+    assert dup_add_term(f, ZZ(-1), 2, ZZ) == dup_normal([1, 1], ZZ)
+
+
+def test_dmp_add_term():
+    assert dmp_add_term([ZZ(1), ZZ(1), ZZ(1)], ZZ(1), 2, 0, ZZ) == \
+        dup_add_term([ZZ(1), ZZ(1), ZZ(1)], ZZ(1), 2, ZZ)
+    assert dmp_add_term(f_0, [[]], 3, 2, ZZ) == f_0
+    assert dmp_add_term(F_0, [[]], 3, 2, QQ) == F_0
+
+
+def test_dup_sub_term():
+    f = dup_normal([], ZZ)
+
+    assert dup_sub_term(f, ZZ(0), 0, ZZ) == dup_normal([], ZZ)
+
+    assert dup_sub_term(f, ZZ(1), 0, ZZ) == dup_normal([-1], ZZ)
+    assert dup_sub_term(f, ZZ(1), 1, ZZ) == dup_normal([-1, 0], ZZ)
+    assert dup_sub_term(f, ZZ(1), 2, ZZ) == dup_normal([-1, 0, 0], ZZ)
+
+    f = dup_normal([1, 1, 1], ZZ)
+
+    assert dup_sub_term(f, ZZ(2), 0, ZZ) == dup_normal([ 1, 1, -1], ZZ)
+    assert dup_sub_term(f, ZZ(2), 1, ZZ) == dup_normal([ 1, -1, 1], ZZ)
+    assert dup_sub_term(f, ZZ(2), 2, ZZ) == dup_normal([-1, 1, 1], ZZ)
+
+    assert dup_sub_term(f, ZZ(1), 3, ZZ) == dup_normal([-1, 1, 1, 1], ZZ)
+    assert dup_sub_term(f, ZZ(1), 4, ZZ) == dup_normal([-1, 0, 1, 1, 1], ZZ)
+    assert dup_sub_term(f, ZZ(1), 5, ZZ) == dup_normal([-1, 0, 0, 1, 1, 1], ZZ)
+    assert dup_sub_term(
+        f, ZZ(1), 6, ZZ) == dup_normal([-1, 0, 0, 0, 1, 1, 1], ZZ)
+
+    assert dup_sub_term(f, ZZ(1), 2, ZZ) == dup_normal([1, 1], ZZ)
+
+
+def test_dmp_sub_term():
+    assert dmp_sub_term([ZZ(1), ZZ(1), ZZ(1)], ZZ(1), 2, 0, ZZ) == \
+        dup_sub_term([ZZ(1), ZZ(1), ZZ(1)], ZZ(1), 2, ZZ)
+    assert dmp_sub_term(f_0, [[]], 3, 2, ZZ) == f_0
+    assert dmp_sub_term(F_0, [[]], 3, 2, QQ) == F_0
+
+
+def test_dup_mul_term():
+    f = dup_normal([], ZZ)
+
+    assert dup_mul_term(f, ZZ(2), 3, ZZ) == dup_normal([], ZZ)
+
+    f = dup_normal([1, 1], ZZ)
+
+    assert dup_mul_term(f, ZZ(0), 3, ZZ) == dup_normal([], ZZ)
+
+    f = dup_normal([1, 2, 3], ZZ)
+
+    assert dup_mul_term(f, ZZ(2), 0, ZZ) == dup_normal([2, 4, 6], ZZ)
+    assert dup_mul_term(f, ZZ(2), 1, ZZ) == dup_normal([2, 4, 6, 0], ZZ)
+    assert dup_mul_term(f, ZZ(2), 2, ZZ) == dup_normal([2, 4, 6, 0, 0], ZZ)
+    assert dup_mul_term(f, ZZ(2), 3, ZZ) == dup_normal([2, 4, 6, 0, 0, 0], ZZ)
+
+
+def test_dmp_mul_term():
+    assert dmp_mul_term([ZZ(1), ZZ(2), ZZ(3)], ZZ(2), 1, 0, ZZ) == \
+        dup_mul_term([ZZ(1), ZZ(2), ZZ(3)], ZZ(2), 1, ZZ)
+
+    assert dmp_mul_term([[]], [ZZ(2)], 3, 1, ZZ) == [[]]
+    assert dmp_mul_term([[ZZ(1)]], [], 3, 1, ZZ) == [[]]
+
+    assert dmp_mul_term([[ZZ(1), ZZ(2)], [ZZ(3)]], [ZZ(2)], 2, 1, ZZ) == \
+        [[ZZ(2), ZZ(4)], [ZZ(6)], [], []]
+
+    assert dmp_mul_term([[]], [QQ(2, 3)], 3, 1, QQ) == [[]]
+    assert dmp_mul_term([[QQ(1, 2)]], [], 3, 1, QQ) == [[]]
+
+    assert dmp_mul_term([[QQ(1, 5), QQ(2, 5)], [QQ(3, 5)]], [QQ(2, 3)], 2, 1, QQ) == \
+        [[QQ(2, 15), QQ(4, 15)], [QQ(6, 15)], [], []]
+
+
+def test_dup_add_ground():
+    f = ZZ.map([1, 2, 3, 4])
+    g = ZZ.map([1, 2, 3, 8])
+
+    assert dup_add_ground(f, ZZ(4), ZZ) == g
+
+
+def test_dmp_add_ground():
+    f = ZZ.map([[1], [2], [3], [4]])
+    g = ZZ.map([[1], [2], [3], [8]])
+
+    assert dmp_add_ground(f, ZZ(4), 1, ZZ) == g
+
+
+def test_dup_sub_ground():
+    f = ZZ.map([1, 2, 3, 4])
+    g = ZZ.map([1, 2, 3, 0])
+
+    assert dup_sub_ground(f, ZZ(4), ZZ) == g
+
+
+def test_dmp_sub_ground():
+    f = ZZ.map([[1], [2], [3], [4]])
+    g = ZZ.map([[1], [2], [3], []])
+
+    assert dmp_sub_ground(f, ZZ(4), 1, ZZ) == g
+
+
+def test_dup_mul_ground():
+    f = dup_normal([], ZZ)
+
+    assert dup_mul_ground(f, ZZ(2), ZZ) == dup_normal([], ZZ)
+
+    f = dup_normal([1, 2, 3], ZZ)
+
+    assert dup_mul_ground(f, ZZ(0), ZZ) == dup_normal([], ZZ)
+    assert dup_mul_ground(f, ZZ(2), ZZ) == dup_normal([2, 4, 6], ZZ)
+
+
+def test_dmp_mul_ground():
+    assert dmp_mul_ground(f_0, ZZ(2), 2, ZZ) == [
+        [[ZZ(2), ZZ(4), ZZ(6)], [ZZ(4)]],
+        [[ZZ(6)]],
+        [[ZZ(8), ZZ(10), ZZ(12)], [ZZ(2), ZZ(4), ZZ(2)], [ZZ(2)]]
+    ]
+
+    assert dmp_mul_ground(F_0, QQ(1, 2), 2, QQ) == [
+        [[QQ(1, 14), QQ(2, 14), QQ(3, 14)], [QQ(2, 14)]],
+        [[QQ(3, 14)]],
+        [[QQ(4, 14), QQ(5, 14), QQ(6, 14)], [QQ(1, 14), QQ(2, 14),
+             QQ(1, 14)], [QQ(1, 14)]]
+    ]
+
+
+def test_dup_quo_ground():
+    raises(ZeroDivisionError, lambda: dup_quo_ground(dup_normal([1, 2,
+           3], ZZ), ZZ(0), ZZ))
+
+    f = dup_normal([], ZZ)
+
+    assert dup_quo_ground(f, ZZ(3), ZZ) == dup_normal([], ZZ)
+
+    f = dup_normal([6, 2, 8], ZZ)
+
+    assert dup_quo_ground(f, ZZ(1), ZZ) == f
+    assert dup_quo_ground(f, ZZ(2), ZZ) == dup_normal([3, 1, 4], ZZ)
+
+    assert dup_quo_ground(f, ZZ(3), ZZ) == dup_normal([2, 0, 2], ZZ)
+
+    f = dup_normal([6, 2, 8], QQ)
+
+    assert dup_quo_ground(f, QQ(1), QQ) == f
+    assert dup_quo_ground(f, QQ(2), QQ) == [QQ(3), QQ(1), QQ(4)]
+    assert dup_quo_ground(f, QQ(7), QQ) == [QQ(6, 7), QQ(2, 7), QQ(8, 7)]
+
+
+def test_dup_exquo_ground():
+    raises(ZeroDivisionError, lambda: dup_exquo_ground(dup_normal([1,
+           2, 3], ZZ), ZZ(0), ZZ))
+    raises(ExactQuotientFailed, lambda: dup_exquo_ground(dup_normal([1,
+           2, 3], ZZ), ZZ(3), ZZ))
+
+    f = dup_normal([], ZZ)
+
+    assert dup_exquo_ground(f, ZZ(3), ZZ) == dup_normal([], ZZ)
+
+    f = dup_normal([6, 2, 8], ZZ)
+
+    assert dup_exquo_ground(f, ZZ(1), ZZ) == f
+    assert dup_exquo_ground(f, ZZ(2), ZZ) == dup_normal([3, 1, 4], ZZ)
+
+    f = dup_normal([6, 2, 8], QQ)
+
+    assert dup_exquo_ground(f, QQ(1), QQ) == f
+    assert dup_exquo_ground(f, QQ(2), QQ) == [QQ(3), QQ(1), QQ(4)]
+    assert dup_exquo_ground(f, QQ(7), QQ) == [QQ(6, 7), QQ(2, 7), QQ(8, 7)]
+
+
+def test_dmp_quo_ground():
+    f = dmp_normal([[6], [2], [8]], 1, ZZ)
+
+    assert dmp_quo_ground(f, ZZ(1), 1, ZZ) == f
+    assert dmp_quo_ground(
+        f, ZZ(2), 1, ZZ) == dmp_normal([[3], [1], [4]], 1, ZZ)
+
+    assert dmp_normal(dmp_quo_ground(
+        f, ZZ(3), 1, ZZ), 1, ZZ) == dmp_normal([[2], [], [2]], 1, ZZ)
+
+
+def test_dmp_exquo_ground():
+    f = dmp_normal([[6], [2], [8]], 1, ZZ)
+
+    assert dmp_exquo_ground(f, ZZ(1), 1, ZZ) == f
+    assert dmp_exquo_ground(
+        f, ZZ(2), 1, ZZ) == dmp_normal([[3], [1], [4]], 1, ZZ)
+
+
+def test_dup_lshift():
+    assert dup_lshift([], 3, ZZ) == []
+    assert dup_lshift([1], 3, ZZ) == [1, 0, 0, 0]
+
+
+def test_dup_rshift():
+    assert dup_rshift([], 3, ZZ) == []
+    assert dup_rshift([1, 0, 0, 0], 3, ZZ) == [1]
+
+
+def test_dup_abs():
+    assert dup_abs([], ZZ) == []
+    assert dup_abs([ZZ( 1)], ZZ) == [ZZ(1)]
+    assert dup_abs([ZZ(-7)], ZZ) == [ZZ(7)]
+    assert dup_abs([ZZ(-1), ZZ(2), ZZ(3)], ZZ) == [ZZ(1), ZZ(2), ZZ(3)]
+
+    assert dup_abs([], QQ) == []
+    assert dup_abs([QQ( 1, 2)], QQ) == [QQ(1, 2)]
+    assert dup_abs([QQ(-7, 3)], QQ) == [QQ(7, 3)]
+    assert dup_abs(
+        [QQ(-1, 7), QQ(2, 7), QQ(3, 7)], QQ) == [QQ(1, 7), QQ(2, 7), QQ(3, 7)]
+
+
+def test_dmp_abs():
+    assert dmp_abs([ZZ(-1)], 0, ZZ) == [ZZ(1)]
+    assert dmp_abs([QQ(-1, 2)], 0, QQ) == [QQ(1, 2)]
+
+    assert dmp_abs([[[]]], 2, ZZ) == [[[]]]
+    assert dmp_abs([[[ZZ(1)]]], 2, ZZ) == [[[ZZ(1)]]]
+    assert dmp_abs([[[ZZ(-7)]]], 2, ZZ) == [[[ZZ(7)]]]
+
+    assert dmp_abs([[[]]], 2, QQ) == [[[]]]
+    assert dmp_abs([[[QQ(1, 2)]]], 2, QQ) == [[[QQ(1, 2)]]]
+    assert dmp_abs([[[QQ(-7, 9)]]], 2, QQ) == [[[QQ(7, 9)]]]
+
+
+def test_dup_neg():
+    assert dup_neg([], ZZ) == []
+    assert dup_neg([ZZ(1)], ZZ) == [ZZ(-1)]
+    assert dup_neg([ZZ(-7)], ZZ) == [ZZ(7)]
+    assert dup_neg([ZZ(-1), ZZ(2), ZZ(3)], ZZ) == [ZZ(1), ZZ(-2), ZZ(-3)]
+
+    assert dup_neg([], QQ) == []
+    assert dup_neg([QQ(1, 2)], QQ) == [QQ(-1, 2)]
+    assert dup_neg([QQ(-7, 9)], QQ) == [QQ(7, 9)]
+    assert dup_neg([QQ(
+        -1, 7), QQ(2, 7), QQ(3, 7)], QQ) == [QQ(1, 7), QQ(-2, 7), QQ(-3, 7)]
+
+
+def test_dmp_neg():
+    assert dmp_neg([ZZ(-1)], 0, ZZ) == [ZZ(1)]
+    assert dmp_neg([QQ(-1, 2)], 0, QQ) == [QQ(1, 2)]
+
+    assert dmp_neg([[[]]], 2, ZZ) == [[[]]]
+    assert dmp_neg([[[ZZ(1)]]], 2, ZZ) == [[[ZZ(-1)]]]
+    assert dmp_neg([[[ZZ(-7)]]], 2, ZZ) == [[[ZZ(7)]]]
+
+    assert dmp_neg([[[]]], 2, QQ) == [[[]]]
+    assert dmp_neg([[[QQ(1, 9)]]], 2, QQ) == [[[QQ(-1, 9)]]]
+    assert dmp_neg([[[QQ(-7, 9)]]], 2, QQ) == [[[QQ(7, 9)]]]
+
+
+def test_dup_add():
+    assert dup_add([], [], ZZ) == []
+    assert dup_add([ZZ(1)], [], ZZ) == [ZZ(1)]
+    assert dup_add([], [ZZ(1)], ZZ) == [ZZ(1)]
+    assert dup_add([ZZ(1)], [ZZ(1)], ZZ) == [ZZ(2)]
+    assert dup_add([ZZ(1)], [ZZ(2)], ZZ) == [ZZ(3)]
+
+    assert dup_add([ZZ(1), ZZ(2)], [ZZ(1)], ZZ) == [ZZ(1), ZZ(3)]
+    assert dup_add([ZZ(1)], [ZZ(1), ZZ(2)], ZZ) == [ZZ(1), ZZ(3)]
+
+    assert dup_add([ZZ(1), ZZ(
+        2), ZZ(3)], [ZZ(8), ZZ(9), ZZ(10)], ZZ) == [ZZ(9), ZZ(11), ZZ(13)]
+
+    assert dup_add([], [], QQ) == []
+    assert dup_add([QQ(1, 2)], [], QQ) == [QQ(1, 2)]
+    assert dup_add([], [QQ(1, 2)], QQ) == [QQ(1, 2)]
+    assert dup_add([QQ(1, 4)], [QQ(1, 4)], QQ) == [QQ(1, 2)]
+    assert dup_add([QQ(1, 4)], [QQ(1, 2)], QQ) == [QQ(3, 4)]
+
+    assert dup_add([QQ(1, 2), QQ(2, 3)], [QQ(1)], QQ) == [QQ(1, 2), QQ(5, 3)]
+    assert dup_add([QQ(1)], [QQ(1, 2), QQ(2, 3)], QQ) == [QQ(1, 2), QQ(5, 3)]
+
+    assert dup_add([QQ(1, 7), QQ(2, 7), QQ(3, 7)], [QQ(
+        8, 7), QQ(9, 7), QQ(10, 7)], QQ) == [QQ(9, 7), QQ(11, 7), QQ(13, 7)]
+
+
+def test_dmp_add():
+    assert dmp_add([ZZ(1), ZZ(2)], [ZZ(1)], 0, ZZ) == \
+        dup_add([ZZ(1), ZZ(2)], [ZZ(1)], ZZ)
+    assert dmp_add([QQ(1, 2), QQ(2, 3)], [QQ(1)], 0, QQ) == \
+        dup_add([QQ(1, 2), QQ(2, 3)], [QQ(1)], QQ)
+
+    assert dmp_add([[[]]], [[[]]], 2, ZZ) == [[[]]]
+    assert dmp_add([[[ZZ(1)]]], [[[]]], 2, ZZ) == [[[ZZ(1)]]]
+    assert dmp_add([[[]]], [[[ZZ(1)]]], 2, ZZ) == [[[ZZ(1)]]]
+    assert dmp_add([[[ZZ(2)]]], [[[ZZ(1)]]], 2, ZZ) == [[[ZZ(3)]]]
+    assert dmp_add([[[ZZ(1)]]], [[[ZZ(2)]]], 2, ZZ) == [[[ZZ(3)]]]
+
+    assert dmp_add([[[]]], [[[]]], 2, QQ) == [[[]]]
+    assert dmp_add([[[QQ(1, 2)]]], [[[]]], 2, QQ) == [[[QQ(1, 2)]]]
+    assert dmp_add([[[]]], [[[QQ(1, 2)]]], 2, QQ) == [[[QQ(1, 2)]]]
+    assert dmp_add([[[QQ(2, 7)]]], [[[QQ(1, 7)]]], 2, QQ) == [[[QQ(3, 7)]]]
+    assert dmp_add([[[QQ(1, 7)]]], [[[QQ(2, 7)]]], 2, QQ) == [[[QQ(3, 7)]]]
+
+
+def test_dup_sub():
+    assert dup_sub([], [], ZZ) == []
+    assert dup_sub([ZZ(1)], [], ZZ) == [ZZ(1)]
+    assert dup_sub([], [ZZ(1)], ZZ) == [ZZ(-1)]
+    assert dup_sub([ZZ(1)], [ZZ(1)], ZZ) == []
+    assert dup_sub([ZZ(1)], [ZZ(2)], ZZ) == [ZZ(-1)]
+
+    assert dup_sub([ZZ(1), ZZ(2)], [ZZ(1)], ZZ) == [ZZ(1), ZZ(1)]
+    assert dup_sub([ZZ(1)], [ZZ(1), ZZ(2)], ZZ) == [ZZ(-1), ZZ(-1)]
+
+    assert dup_sub([ZZ(3), ZZ(
+        2), ZZ(1)], [ZZ(8), ZZ(9), ZZ(10)], ZZ) == [ZZ(-5), ZZ(-7), ZZ(-9)]
+
+    assert dup_sub([], [], QQ) == []
+    assert dup_sub([QQ(1, 2)], [], QQ) == [QQ(1, 2)]
+    assert dup_sub([], [QQ(1, 2)], QQ) == [QQ(-1, 2)]
+    assert dup_sub([QQ(1, 3)], [QQ(1, 3)], QQ) == []
+    assert dup_sub([QQ(1, 3)], [QQ(2, 3)], QQ) == [QQ(-1, 3)]
+
+    assert dup_sub([QQ(1, 7), QQ(2, 7)], [QQ(1)], QQ) == [QQ(1, 7), QQ(-5, 7)]
+    assert dup_sub([QQ(1)], [QQ(1, 7), QQ(2, 7)], QQ) == [QQ(-1, 7), QQ(5, 7)]
+
+    assert dup_sub([QQ(3, 7), QQ(2, 7), QQ(1, 7)], [QQ(
+        8, 7), QQ(9, 7), QQ(10, 7)], QQ) == [QQ(-5, 7), QQ(-7, 7), QQ(-9, 7)]
+
+
+def test_dmp_sub():
+    assert dmp_sub([ZZ(1), ZZ(2)], [ZZ(1)], 0, ZZ) == \
+        dup_sub([ZZ(1), ZZ(2)], [ZZ(1)], ZZ)
+    assert dmp_sub([QQ(1, 2), QQ(2, 3)], [QQ(1)], 0, QQ) == \
+        dup_sub([QQ(1, 2), QQ(2, 3)], [QQ(1)], QQ)
+
+    assert dmp_sub([[[]]], [[[]]], 2, ZZ) == [[[]]]
+    assert dmp_sub([[[ZZ(1)]]], [[[]]], 2, ZZ) == [[[ZZ(1)]]]
+    assert dmp_sub([[[]]], [[[ZZ(1)]]], 2, ZZ) == [[[ZZ(-1)]]]
+    assert dmp_sub([[[ZZ(2)]]], [[[ZZ(1)]]], 2, ZZ) == [[[ZZ(1)]]]
+    assert dmp_sub([[[ZZ(1)]]], [[[ZZ(2)]]], 2, ZZ) == [[[ZZ(-1)]]]
+
+    assert dmp_sub([[[]]], [[[]]], 2, QQ) == [[[]]]
+    assert dmp_sub([[[QQ(1, 2)]]], [[[]]], 2, QQ) == [[[QQ(1, 2)]]]
+    assert dmp_sub([[[]]], [[[QQ(1, 2)]]], 2, QQ) == [[[QQ(-1, 2)]]]
+    assert dmp_sub([[[QQ(2, 7)]]], [[[QQ(1, 7)]]], 2, QQ) == [[[QQ(1, 7)]]]
+    assert dmp_sub([[[QQ(1, 7)]]], [[[QQ(2, 7)]]], 2, QQ) == [[[QQ(-1, 7)]]]
+
+
+def test_dup_add_mul():
+    assert dup_add_mul([ZZ(1), ZZ(2), ZZ(3)], [ZZ(3), ZZ(2), ZZ(1)],
+               [ZZ(1), ZZ(2)], ZZ) == [ZZ(3), ZZ(9), ZZ(7), ZZ(5)]
+    assert dmp_add_mul([[ZZ(1), ZZ(2)], [ZZ(3)]], [[ZZ(3)], [ZZ(2), ZZ(1)]],
+               [[ZZ(1)], [ZZ(2)]], 1, ZZ) == [[ZZ(3)], [ZZ(3), ZZ(9)], [ZZ(4), ZZ(5)]]
+
+
+def test_dup_sub_mul():
+    assert dup_sub_mul([ZZ(1), ZZ(2), ZZ(3)], [ZZ(3), ZZ(2), ZZ(1)],
+               [ZZ(1), ZZ(2)], ZZ) == [ZZ(-3), ZZ(-7), ZZ(-3), ZZ(1)]
+    assert dmp_sub_mul([[ZZ(1), ZZ(2)], [ZZ(3)]], [[ZZ(3)], [ZZ(2), ZZ(1)]],
+               [[ZZ(1)], [ZZ(2)]], 1, ZZ) == [[ZZ(-3)], [ZZ(-1), ZZ(-5)], [ZZ(-4), ZZ(1)]]
+
+
+def test_dup_mul():
+    assert dup_mul([], [], ZZ) == []
+    assert dup_mul([], [ZZ(1)], ZZ) == []
+    assert dup_mul([ZZ(1)], [], ZZ) == []
+    assert dup_mul([ZZ(1)], [ZZ(1)], ZZ) == [ZZ(1)]
+    assert dup_mul([ZZ(5)], [ZZ(7)], ZZ) == [ZZ(35)]
+
+    assert dup_mul([], [], QQ) == []
+    assert dup_mul([], [QQ(1, 2)], QQ) == []
+    assert dup_mul([QQ(1, 2)], [], QQ) == []
+    assert dup_mul([QQ(1, 2)], [QQ(4, 7)], QQ) == [QQ(2, 7)]
+    assert dup_mul([QQ(5, 7)], [QQ(3, 7)], QQ) == [QQ(15, 49)]
+
+    f = dup_normal([3, 0, 0, 6, 1, 2], ZZ)
+    g = dup_normal([4, 0, 1, 0], ZZ)
+    h = dup_normal([12, 0, 3, 24, 4, 14, 1, 2, 0], ZZ)
+
+    assert dup_mul(f, g, ZZ) == h
+    assert dup_mul(g, f, ZZ) == h
+
+    f = dup_normal([2, 0, 0, 1, 7], ZZ)
+    h = dup_normal([4, 0, 0, 4, 28, 0, 1, 14, 49], ZZ)
+
+    assert dup_mul(f, f, ZZ) == h
+
+    K = FF(6)
+
+    assert dup_mul([K(2), K(1)], [K(3), K(4)], K) == [K(5), K(4)]
+
+    p1 = dup_normal([79, -1, 78, -94, -10, 11, 32, -19, 78, 2, -89, 30, 73, 42,
+        85, 77, 83, -30, -34, -2, 95, -81, 37, -49, -46, -58, -16, 37, 35, -11,
+        -57, -15, -31, 67, -20, 27, 76, 2, 70, 67, -65, 65, -26, -93, -44, -12,
+        -92, 57, -90, -57, -11, -67, -98, -69, 97, -41, 89, 33, 89, -50, 81,
+        -31, 60, -27, 43, 29, -77, 44, 21, -91, 32, -57, 33, 3, 53, -51, -38,
+        -99, -84, 23, -50, 66, -100, 1, -75, -25, 27, -60, 98, -51, -87, 6, 8,
+        78, -28, -95, -88, 12, -35, 26, -9, 16, -92, 55, -7, -86, 68, -39, -46,
+        84, 94, 45, 60, 92, 68, -75, -74, -19, 8, 75, 78, 91, 57, 34, 14, -3,
+        -49, 65, 78, -18, 6, -29, -80, -98, 17, 13, 58, 21, 20, 9, 37, 7, -30,
+        -53, -20, 34, 67, -42, 89, -22, 73, 43, -6, 5, 51, -8, -15, -52, -22,
+        -58, -72, -3, 43, -92, 82, 83, -2, -13, -23, -60, 16, -94, -8, -28,
+        -95, -72, 63, -90, 76, 6, -43, -100, -59, 76, 3, 3, 46, -85, 75, 62,
+        -71, -76, 88, 97, -72, -1, 30, -64, 72, -48, 14, -78, 58, 63, -91, 24,
+        -87, -27, -80, -100, -44, 98, 70, 100, -29, -38, 11, 77, 100, 52, 86,
+        65, -5, -42, -81, -38, -42, 43, -2, -70, -63, -52], ZZ)
+    p2 = dup_normal([65, -19, -47, 1, 90, 81, -15, -34, 25, -75, 9, -83, 50, -5,
+        -44, 31, 1, 70, -7, 78, 74, 80, 85, 65, 21, 41, 66, 19, -40, 63, -21,
+        -27, 32, 69, 83, 34, -35, 14, 81, 57, -75, 32, -67, -89, -100, -61, 46,
+        84, -78, -29, -50, -94, -24, -32, -68, -16, 100, -7, -72, -89, 35, 82,
+        58, 81, -92, 62, 5, -47, -39, -58, -72, -13, 84, 44, 55, -25, 48, -54,
+        -31, -56, -11, -50, -84, 10, 67, 17, 13, -14, 61, 76, -64, -44, -40,
+        -96, 11, -11, -94, 2, 6, 27, -6, 68, -54, 66, -74, -14, -1, -24, -73,
+        96, 89, -11, -89, 56, -53, 72, -43, 96, 25, 63, -31, 29, 68, 83, 91,
+        -93, -19, -38, -40, 40, -12, -19, -79, 44, 100, -66, -29, -77, 62, 39,
+        -8, 11, -97, 14, 87, 64, 21, -18, 13, 15, -59, -75, -99, -88, 57, 54,
+        56, -67, 6, -63, -59, -14, 28, 87, -20, -39, 84, -91, -2, 49, -75, 11,
+        -24, -95, 36, 66, 5, 25, -72, -40, 86, 90, 37, -33, 57, -35, 29, -18,
+        4, -79, 64, -17, -27, 21, 29, -5, -44, -87, -24, 52, 78, 11, -23, -53,
+        36, 42, 21, -68, 94, -91, -51, -21, 51, -76, 72, 31, 24, -48, -80, -9,
+        37, -47, -6, -8, -63, -91, 79, -79, -100, 38, -20, 38, 100, 83, -90,
+        87, 63, -36, 82, -19, 18, -98, -38, 26, 98, -70, 79, 92, 12, 12, 70,
+        74, 36, 48, -13, 31, 31, -47, -71, -12, -64, 36, -42, 32, -86, 60, 83,
+        70, 55, 0, 1, 29, -35, 8, -82, 8, -73, -46, -50, 43, 48, -5, -86, -72,
+        44, -90, 19, 19, 5, -20, 97, -13, -66, -5, 5, -69, 64, -30, 41, 51, 36,
+        13, -99, -61, 94, -12, 74, 98, 68, 24, 46, -97, -87, -6, -27, 82, 62,
+        -11, -77, 86, 66, -47, -49, -50, 13, 18, 89, -89, 46, -80, 13, 98, -35,
+        -36, -25, 12, 20, 26, -52, 79, 27, 79, 100, 8, 62, -58, -28, 37], ZZ)
+    res = dup_normal([5135, -1566, 1376, -7466, 4579, 11710, 8001, -7183,
+        -3737, -7439, 345, -10084, 24522, -1201, 1070, -10245, 9582, 9264,
+        1903, 23312, 18953, 10037, -15268, -5450, 6442, -6243, -3777, 5110,
+        10936, -16649, -6022, 16255, 31300, 24818, 31922, 32760, 7854, 27080,
+        15766, 29596, 7139, 31945, -19810, 465, -38026, -3971, 9641, 465,
+        -19375, 5524, -30112, -11960, -12813, 13535, 30670, 5925, -43725,
+        -14089, 11503, -22782, 6371, 43881, 37465, -33529, -33590, -39798,
+        -37854, -18466, -7908, -35825, -26020, -36923, -11332, -5699, 25166,
+        -3147, 19885, 12962, -20659, -1642, 27723, -56331, -24580, -11010,
+        -20206, 20087, -23772, -16038, 38580, 20901, -50731, 32037, -4299,
+        26508, 18038, -28357, 31846, -7405, -20172, -15894, 2096, 25110,
+        -45786, 45918, -55333, -31928, -49428, -29824, -58796, -24609, -15408,
+        69, -35415, -18439, 10123, -20360, -65949, 33356, -20333, 26476,
+        -32073, 33621, 930, 28803, -42791, 44716, 38164, 12302, -1739, 11421,
+        73385, -7613, 14297, 38155, -414, 77587, 24338, -21415, 29367, 42639,
+        13901, -288, 51027, -11827, 91260, 43407, 88521, -15186, 70572, -12049,
+        5090, -12208, -56374, 15520, -623, -7742, 50825, 11199, -14894, 40892,
+        59591, -31356, -28696, -57842, -87751, -33744, -28436, -28945, -40287,
+        37957, -35638, 33401, -61534, 14870, 40292, 70366, -10803, 102290,
+        -71719, -85251, 7902, -22409, 75009, 99927, 35298, -1175, -762, -34744,
+        -10587, -47574, -62629, -19581, -43659, -54369, -32250, -39545, 15225,
+        -24454, 11241, -67308, -30148, 39929, 37639, 14383, -73475, -77636,
+        -81048, -35992, 41601, -90143, 76937, -8112, 56588, 9124, -40094,
+        -32340, 13253, 10898, -51639, 36390, 12086, -1885, 100714, -28561,
+        -23784, -18735, 18916, 16286, 10742, -87360, -13697, 10689, -19477,
+        -29770, 5060, 20189, -8297, 112407, 47071, 47743, 45519, -4109, 17468,
+        -68831, 78325, -6481, -21641, -19459, 30919, 96115, 8607, 53341, 32105,
+        -16211, 23538, 57259, -76272, -40583, 62093, 38511, -34255, -40665,
+        -40604, -37606, -15274, 33156, -13885, 103636, 118678, -14101, -92682,
+        -100791, 2634, 63791, 98266, 19286, -34590, -21067, -71130, 25380,
+        -40839, -27614, -26060, 52358, -15537, 27138, -6749, 36269, -33306,
+        13207, -91084, -5540, -57116, 69548, 44169, -57742, -41234, -103327,
+        -62904, -8566, 41149, -12866, 71188, 23980, 1838, 58230, 73950, 5594,
+        43113, -8159, -15925, 6911, 85598, -75016, -16214, -62726, -39016,
+        8618, -63882, -4299, 23182, 49959, 49342, -3238, -24913, -37138, 78361,
+        32451, 6337, -11438, -36241, -37737, 8169, -3077, -24829, 57953, 53016,
+        -31511, -91168, 12599, -41849, 41576, 55275, -62539, 47814, -62319,
+        12300, -32076, -55137, -84881, -27546, 4312, -3433, -54382, 113288,
+        -30157, 74469, 18219, 79880, -2124, 98911, 17655, -33499, -32861,
+        47242, -37393, 99765, 14831, -44483, 10800, -31617, -52710, 37406,
+        22105, 29704, -20050, 13778, 43683, 36628, 8494, 60964, -22644, 31550,
+        -17693, 33805, -124879, -12302, 19343, 20400, -30937, -21574, -34037,
+        -33380, 56539, -24993, -75513, -1527, 53563, 65407, -101, 53577, 37991,
+        18717, -23795, -8090, -47987, -94717, 41967, 5170, -14815, -94311,
+        17896, -17734, -57718, -774, -38410, 24830, 29682, 76480, 58802,
+        -46416, -20348, -61353, -68225, -68306, 23822, -31598, 42972, 36327,
+        28968, -65638, -21638, 24354, -8356, 26777, 52982, -11783, -44051,
+        -26467, -44721, -28435, -53265, -25574, -2669, 44155, 22946, -18454,
+        -30718, -11252, 58420, 8711, 67447, 4425, 41749, 67543, 43162, 11793,
+        -41907, 20477, -13080, 6559, -6104, -13244, 42853, 42935, 29793, 36730,
+        -28087, 28657, 17946, 7503, 7204, 21491, -27450, -24241, -98156,
+        -18082, -42613, -24928, 10775, -14842, -44127, 55910, 14777, 31151, -2194,
+        39206, -2100, -4211, 11827, -8918, -19471, 72567, 36447, -65590, -34861,
+        -17147, -45303, 9025, -7333, -35473, 11101, 11638, 3441, 6626, -41800,
+        9416, 13679, 33508, 40502, -60542, 16358, 8392, -43242, -35864, -34127,
+        -48721, 35878, 30598, 28630, 20279, -19983, -14638, -24455, -1851, -11344,
+        45150, 42051, 26034, -28889, -32382, -3527, -14532, 22564, -22346, 477,
+        11706, 28338, -25972, -9185, -22867, -12522, 32120, -4424, 11339, -33913,
+        -7184, 5101, -23552, -17115, -31401, -6104, 21906, 25708, 8406, 6317,
+        -7525, 5014, 20750, 20179, 22724, 11692, 13297, 2493, -253, -16841, -17339,
+        -6753, -4808, 2976, -10881, -10228, -13816, -12686, 1385, 2316, 2190, -875,
+        -1924], ZZ)
+
+    assert dup_mul(p1, p2, ZZ) == res
+
+    p1 = dup_normal([83, -61, -86, -24, 12, 43, -88, -9, 42, 55, -66, 74, 95,
+        -25, -12, 68, -99, 4, 45, 6, -15, -19, 78, 65, -55, 47, -13, 17, 86,
+        81, -58, -27, 50, -40, -24, 39, -41, -92, 75, 90, -1, 40, -15, -27,
+        -35, 68, 70, -64, -40, 78, -88, -58, -39, 69, 46, 12, 28, -94, -37,
+        -50, -80, -96, -61, 25, 1, 71, 4, 12, 48, 4, 34, -47, -75, 5, 48, 82,
+        88, 23, 98, 35, 17, -10, 48, -61, -95, 47, 65, -19, -66, -57, -6, -51,
+        -42, -89, 66, -13, 18, 37, 90, -23, 72, 96, -53, 0, 40, -73, -52, -68,
+        32, -25, -53, 79, -52, 18, 44, 73, -81, 31, -90, 70, 3, 36, 48, 76,
+        -24, -44, 23, 98, -4, 73, 69, 88, -70, 14, -68, 94, -78, -15, -64, -97,
+        -70, -35, 65, 88, 49, -53, -7, 12, -45, -7, 59, -94, 99, -2, 67, -60,
+        -71, 29, -62, -77, 1, 51, 17, 80, -20, -47, -19, 24, -9, 39, -23, 21,
+        -84, 10, 84, 56, -17, -21, -66, 85, 70, 46, -51, -22, -95, 78, -60,
+        -96, -97, -45, 72, 35, 30, -61, -92, -93, -60, -61, 4, -4, -81, -73,
+        46, 53, -11, 26, 94, 45, 14, -78, 55, 84, -68, 98, 60, 23, 100, -63,
+        68, 96, -16, 3, 56, 21, -58, 62, -67, 66, 85, 41, -79, -22, 97, -67,
+        82, 82, -96, -20, -7, 48, -67, 48, -9, -39, 78], ZZ)
+    p2 = dup_normal([52, 88, 76, 66, 9, -64, 46, -20, -28, 69, 60, 96, -36,
+        -92, -30, -11, -35, 35, 55, 63, -92, -7, 25, -58, 74, 55, -6, 4, 47,
+        -92, -65, 67, -45, 74, -76, 59, -6, 69, 39, 24, -71, -7, 39, -45, 60,
+        -68, 98, 97, -79, 17, 4, 94, -64, 68, -100, -96, -2, 3, 22, 96, 54,
+        -77, -86, 67, 6, 57, 37, 40, 89, -78, 64, -94, -45, -92, 57, 87, -26,
+        36, 19, 97, 25, 77, -87, 24, 43, -5, 35, 57, 83, 71, 35, 63, 61, 96,
+        -22, 8, -1, 96, 43, 45, 94, -93, 36, 71, -41, -99, 85, -48, 59, 52,
+        -17, 5, 87, -16, -68, -54, 76, -18, 100, 91, -42, -70, -66, -88, -12,
+        1, 95, -82, 52, 43, -29, 3, 12, 72, -99, -43, -32, -93, -51, 16, -20,
+        -12, -11, 5, 33, -38, 93, -5, -74, 25, 74, -58, 93, 59, -63, -86, 63,
+        -20, -4, -74, -73, -95, 29, -28, 93, -91, -2, -38, -62, 77, -58, -85,
+        -28, 95, 38, 19, -69, 86, 94, 25, -2, -4, 47, 34, -59, 35, -48, 29,
+        -63, -53, 34, 29, 66, 73, 6, 92, -84, 89, 15, 81, 93, 97, 51, -72, -78,
+        25, 60, 90, -45, 39, 67, -84, -62, 57, 26, -32, -56, -14, -83, 76, 5,
+        -2, 99, -100, 28, 46, 94, -7, 53, -25, 16, -23, -36, 89, -78, -63, 31,
+        1, 84, -99, -52, 76, 48, 90, -76, 44, -19, 54, -36, -9, -73, -100, -69,
+        31, 42, 25, -39, 76, -26, -8, -14, 51, 3, 37, 45, 2, -54, 13, -34, -92,
+        17, -25, -65, 53, -63, 30, 4, -70, -67, 90, 52, 51, 18, -3, 31, -45,
+        -9, 59, 63, -87, 22, -32, 29, -38, 21, 36, -82, 27, -11], ZZ)
+    res = dup_normal([4316, 4132, -3532, -7974, -11303, -10069, 5484, -3330,
+        -5874, 7734, 4673, 11327, -9884, -8031, 17343, 21035, -10570, -9285,
+        15893, 3780, -14083, 8819, 17592, 10159, 7174, -11587, 8598, -16479,
+        3602, 25596, 9781, 12163, 150, 18749, -21782, -12307, 27578, -2757,
+        -12573, 12565, 6345, -18956, 19503, -15617, 1443, -16778, 36851, 23588,
+        -28474, 5749, 40695, -7521, -53669, -2497, -18530, 6770, 57038, 3926,
+        -6927, -15399, 1848, -64649, -27728, 3644, 49608, 15187, -8902, -9480,
+        -7398, -40425, 4824, 23767, -7594, -6905, 33089, 18786, 12192, 24670,
+        31114, 35334, -4501, -14676, 7107, -59018, -21352, 20777, 19661, 20653,
+        33754, -885, -43758, 6269, 51897, -28719, -97488, -9527, 13746, 11644,
+        17644, -21720, 23782, -10481, 47867, 20752, 33810, -1875, 39918, -7710,
+        -40840, 19808, -47075, 23066, 46616, 25201, 9287, 35436, -1602, 9645,
+        -11978, 13273, 15544, 33465, 20063, 44539, 11687, 27314, -6538, -37467,
+        14031, 32970, -27086, 41323, 29551, 65910, -39027, -37800, -22232,
+        8212, 46316, -28981, -55282, 50417, -44929, -44062, 73879, 37573,
+        -2596, -10877, -21893, -133218, -33707, -25753, -9531, 17530, 61126,
+        2748, -56235, 43874, -10872, -90459, -30387, 115267, -7264, -44452,
+        122626, 14839, -599, 10337, 57166, -67467, -54957, 63669, 1202, 18488,
+        52594, 7205, -97822, 612, 78069, -5403, -63562, 47236, 36873, -154827,
+        -26188, 82427, -39521, 5628, 7416, 5276, -53095, 47050, 26121, -42207,
+        79021, -13035, 2499, -66943, 29040, -72355, -23480, 23416, -12885,
+        -44225, -42688, -4224, 19858, 55299, 15735, 11465, 101876, -39169,
+        51786, 14723, 43280, -68697, 16410, 92295, 56767, 7183, 111850, 4550,
+        115451, -38443, -19642, -35058, 10230, 93829, 8925, 63047, 3146, 29250,
+        8530, 5255, -98117, -115517, -76817, -8724, 41044, 1312, -35974, 79333,
+        -28567, 7547, -10580, -24559, -16238, 10794, -3867, 24848, 57770,
+        -51536, -35040, 71033, 29853, 62029, -7125, -125585, -32169, -47907,
+        156811, -65176, -58006, -15757, -57861, 11963, 30225, -41901, -41681,
+        31310, 27982, 18613, 61760, 60746, -59096, 33499, 30097, -17997, 24032,
+        56442, -83042, 23747, -20931, -21978, -158752, -9883, -73598, -7987,
+        -7333, -125403, -116329, 30585, 53281, 51018, -29193, 88575, 8264,
+        -40147, -16289, 113088, 12810, -6508, 101552, -13037, 34440, -41840,
+        101643, 24263, 80532, 61748, 65574, 6423, -20672, 6591, -10834, -71716,
+        86919, -92626, 39161, 28490, 81319, 46676, 106720, 43530, 26998, 57456,
+        -8862, 60989, 13982, 3119, -2224, 14743, 55415, -49093, -29303, 28999,
+        1789, 55953, -84043, -7780, -65013, 57129, -47251, 61484, 61994,
+        -78361, -82778, 22487, -26894, 9756, -74637, -15519, -4360, 30115,
+        42433, 35475, 15286, 69768, 21509, -20214, 78675, -21163, 13596, 11443,
+        -10698, -53621, -53867, -24155, 64500, -42784, -33077, -16500, 873,
+        -52788, 14546, -38011, 36974, -39849, -34029, -94311, 83068, -50437,
+        -26169, -46746, 59185, 42259, -101379, -12943, 30089, -59086, 36271,
+        22723, -30253, -52472, -70826, -23289, 3331, -31687, 14183, -857,
+        -28627, 35246, -51284, 5636, -6933, 66539, 36654, 50927, 24783, 3457,
+        33276, 45281, 45650, -4938, -9968, -22590, 47995, 69229, 5214, -58365,
+        -17907, -14651, 18668, 18009, 12649, -11851, -13387, 20339, 52472,
+        -1087, -21458, -68647, 52295, 15849, 40608, 15323, 25164, -29368,
+        10352, -7055, 7159, 21695, -5373, -54849, 101103, -24963, -10511,
+        33227, 7659, 41042, -69588, 26718, -20515, 6441, 38135, -63, 24088,
+        -35364, -12785, -18709, 47843, 48533, -48575, 17251, -19394, 32878,
+        -9010, -9050, 504, -12407, 28076, -3429, 25324, -4210, -26119, 752,
+        -29203, 28251, -11324, -32140, -3366, -25135, 18702, -31588, -7047,
+        -24267, 49987, -14975, -33169, 37744, -7720, -9035, 16964, -2807, -421,
+        14114, -17097, -13662, 40628, -12139, -9427, 5369, 17551, -13232, -16211,
+        9804, -7422, 2677, 28635, -8280, -4906, 2908, -22558, 5604, 12459, 8756,
+        -3980, -4745, -18525, 7913, 5970, -16457, 20230, -6247, -13812, 2505,
+        11899, 1409, -15094, 22540, -18863, 137, 11123, -4516, 2290, -8594, 12150,
+        -10380, 3005, 5235, -7350, 2535, -858], ZZ)
+
+    assert dup_mul(p1, p2, ZZ) == res
+
+
+def test_dmp_mul():
+    assert dmp_mul([ZZ(5)], [ZZ(7)], 0, ZZ) == \
+        dup_mul([ZZ(5)], [ZZ(7)], ZZ)
+    assert dmp_mul([QQ(5, 7)], [QQ(3, 7)], 0, QQ) == \
+        dup_mul([QQ(5, 7)], [QQ(3, 7)], QQ)
+
+    assert dmp_mul([[[]]], [[[]]], 2, ZZ) == [[[]]]
+    assert dmp_mul([[[ZZ(1)]]], [[[]]], 2, ZZ) == [[[]]]
+    assert dmp_mul([[[]]], [[[ZZ(1)]]], 2, ZZ) == [[[]]]
+    assert dmp_mul([[[ZZ(2)]]], [[[ZZ(1)]]], 2, ZZ) == [[[ZZ(2)]]]
+    assert dmp_mul([[[ZZ(1)]]], [[[ZZ(2)]]], 2, ZZ) == [[[ZZ(2)]]]
+
+    assert dmp_mul([[[]]], [[[]]], 2, QQ) == [[[]]]
+    assert dmp_mul([[[QQ(1, 2)]]], [[[]]], 2, QQ) == [[[]]]
+    assert dmp_mul([[[]]], [[[QQ(1, 2)]]], 2, QQ) == [[[]]]
+    assert dmp_mul([[[QQ(2, 7)]]], [[[QQ(1, 3)]]], 2, QQ) == [[[QQ(2, 21)]]]
+    assert dmp_mul([[[QQ(1, 7)]]], [[[QQ(2, 3)]]], 2, QQ) == [[[QQ(2, 21)]]]
+
+    K = FF(6)
+
+    assert dmp_mul(
+        [[K(2)], [K(1)]], [[K(3)], [K(4)]], 1, K) == [[K(5)], [K(4)]]
+
+
+def test_dup_sqr():
+    assert dup_sqr([], ZZ) == []
+    assert dup_sqr([ZZ(2)], ZZ) == [ZZ(4)]
+    assert dup_sqr([ZZ(1), ZZ(2)], ZZ) == [ZZ(1), ZZ(4), ZZ(4)]
+
+    assert dup_sqr([], QQ) == []
+    assert dup_sqr([QQ(2, 3)], QQ) == [QQ(4, 9)]
+    assert dup_sqr([QQ(1, 3), QQ(2, 3)], QQ) == [QQ(1, 9), QQ(4, 9), QQ(4, 9)]
+
+    f = dup_normal([2, 0, 0, 1, 7], ZZ)
+
+    assert dup_sqr(f, ZZ) == dup_normal([4, 0, 0, 4, 28, 0, 1, 14, 49], ZZ)
+
+    K = FF(9)
+
+    assert dup_sqr([K(3), K(4)], K) == [K(6), K(7)]
+
+
+def test_dmp_sqr():
+    assert dmp_sqr([ZZ(1), ZZ(2)], 0, ZZ) == \
+        dup_sqr([ZZ(1), ZZ(2)], ZZ)
+
+    assert dmp_sqr([[[]]], 2, ZZ) == [[[]]]
+    assert dmp_sqr([[[ZZ(2)]]], 2, ZZ) == [[[ZZ(4)]]]
+
+    assert dmp_sqr([[[]]], 2, QQ) == [[[]]]
+    assert dmp_sqr([[[QQ(2, 3)]]], 2, QQ) == [[[QQ(4, 9)]]]
+
+    K = FF(9)
+
+    assert dmp_sqr([[K(3)], [K(4)]], 1, K) == [[K(6)], [K(7)]]
+
+
+def test_dup_pow():
+    assert dup_pow([], 0, ZZ) == [ZZ(1)]
+    assert dup_pow([], 0, QQ) == [QQ(1)]
+
+    assert dup_pow([], 1, ZZ) == []
+    assert dup_pow([], 7, ZZ) == []
+
+    assert dup_pow([ZZ(1)], 0, ZZ) == [ZZ(1)]
+    assert dup_pow([ZZ(1)], 1, ZZ) == [ZZ(1)]
+    assert dup_pow([ZZ(1)], 7, ZZ) == [ZZ(1)]
+
+    assert dup_pow([ZZ(3)], 0, ZZ) == [ZZ(1)]
+    assert dup_pow([ZZ(3)], 1, ZZ) == [ZZ(3)]
+    assert dup_pow([ZZ(3)], 7, ZZ) == [ZZ(2187)]
+
+    assert dup_pow([QQ(1, 1)], 0, QQ) == [QQ(1, 1)]
+    assert dup_pow([QQ(1, 1)], 1, QQ) == [QQ(1, 1)]
+    assert dup_pow([QQ(1, 1)], 7, QQ) == [QQ(1, 1)]
+
+    assert dup_pow([QQ(3, 7)], 0, QQ) == [QQ(1, 1)]
+    assert dup_pow([QQ(3, 7)], 1, QQ) == [QQ(3, 7)]
+    assert dup_pow([QQ(3, 7)], 7, QQ) == [QQ(2187, 823543)]
+
+    f = dup_normal([2, 0, 0, 1, 7], ZZ)
+
+    assert dup_pow(f, 0, ZZ) == dup_normal([1], ZZ)
+    assert dup_pow(f, 1, ZZ) == dup_normal([2, 0, 0, 1, 7], ZZ)
+    assert dup_pow(f, 2, ZZ) == dup_normal([4, 0, 0, 4, 28, 0, 1, 14, 49], ZZ)
+    assert dup_pow(f, 3, ZZ) == dup_normal(
+        [8, 0, 0, 12, 84, 0, 6, 84, 294, 1, 21, 147, 343], ZZ)
+
+
+def test_dmp_pow():
+    assert dmp_pow([[]], 0, 1, ZZ) == [[ZZ(1)]]
+    assert dmp_pow([[]], 0, 1, QQ) == [[QQ(1)]]
+
+    assert dmp_pow([[]], 1, 1, ZZ) == [[]]
+    assert dmp_pow([[]], 7, 1, ZZ) == [[]]
+
+    assert dmp_pow([[ZZ(1)]], 0, 1, ZZ) == [[ZZ(1)]]
+    assert dmp_pow([[ZZ(1)]], 1, 1, ZZ) == [[ZZ(1)]]
+    assert dmp_pow([[ZZ(1)]], 7, 1, ZZ) == [[ZZ(1)]]
+
+    assert dmp_pow([[QQ(3, 7)]], 0, 1, QQ) == [[QQ(1, 1)]]
+    assert dmp_pow([[QQ(3, 7)]], 1, 1, QQ) == [[QQ(3, 7)]]
+    assert dmp_pow([[QQ(3, 7)]], 7, 1, QQ) == [[QQ(2187, 823543)]]
+
+    f = dup_normal([2, 0, 0, 1, 7], ZZ)
+
+    assert dmp_pow(f, 2, 0, ZZ) == dup_pow(f, 2, ZZ)
+
+
+def test_dup_pdiv():
+    f = dup_normal([3, 1, 1, 5], ZZ)
+    g = dup_normal([5, -3, 1], ZZ)
+
+    q = dup_normal([15, 14], ZZ)
+    r = dup_normal([52, 111], ZZ)
+
+    assert dup_pdiv(f, g, ZZ) == (q, r)
+    assert dup_pquo(f, g, ZZ) == q
+    assert dup_prem(f, g, ZZ) == r
+
+    raises(ExactQuotientFailed, lambda: dup_pexquo(f, g, ZZ))
+
+    f = dup_normal([3, 1, 1, 5], QQ)
+    g = dup_normal([5, -3, 1], QQ)
+
+    q = dup_normal([15, 14], QQ)
+    r = dup_normal([52, 111], QQ)
+
+    assert dup_pdiv(f, g, QQ) == (q, r)
+    assert dup_pquo(f, g, QQ) == q
+    assert dup_prem(f, g, QQ) == r
+
+    raises(ExactQuotientFailed, lambda: dup_pexquo(f, g, QQ))
+
+
+def test_dmp_pdiv():
+    f = dmp_normal([[1], [], [1, 0, 0]], 1, ZZ)
+    g = dmp_normal([[1], [-1, 0]], 1, ZZ)
+
+    q = dmp_normal([[1], [1, 0]], 1, ZZ)
+    r = dmp_normal([[2, 0, 0]], 1, ZZ)
+
+    assert dmp_pdiv(f, g, 1, ZZ) == (q, r)
+    assert dmp_pquo(f, g, 1, ZZ) == q
+    assert dmp_prem(f, g, 1, ZZ) == r
+
+    raises(ExactQuotientFailed, lambda: dmp_pexquo(f, g, 1, ZZ))
+
+    f = dmp_normal([[1], [], [1, 0, 0]], 1, ZZ)
+    g = dmp_normal([[2], [-2, 0]], 1, ZZ)
+
+    q = dmp_normal([[2], [2, 0]], 1, ZZ)
+    r = dmp_normal([[8, 0, 0]], 1, ZZ)
+
+    assert dmp_pdiv(f, g, 1, ZZ) == (q, r)
+    assert dmp_pquo(f, g, 1, ZZ) == q
+    assert dmp_prem(f, g, 1, ZZ) == r
+
+    raises(ExactQuotientFailed, lambda: dmp_pexquo(f, g, 1, ZZ))
+
+
+def test_dup_rr_div():
+    raises(ZeroDivisionError, lambda: dup_rr_div([1, 2, 3], [], ZZ))
+
+    f = dup_normal([3, 1, 1, 5], ZZ)
+    g = dup_normal([5, -3, 1], ZZ)
+
+    q, r = [], f
+
+    assert dup_rr_div(f, g, ZZ) == (q, r)
+
+
+def test_dmp_rr_div():
+    raises(ZeroDivisionError, lambda: dmp_rr_div([[1, 2], [3]], [[]], 1, ZZ))
+
+    f = dmp_normal([[1], [], [1, 0, 0]], 1, ZZ)
+    g = dmp_normal([[1], [-1, 0]], 1, ZZ)
+
+    q = dmp_normal([[1], [1, 0]], 1, ZZ)
+    r = dmp_normal([[2, 0, 0]], 1, ZZ)
+
+    assert dmp_rr_div(f, g, 1, ZZ) == (q, r)
+
+    f = dmp_normal([[1], [], [1, 0, 0]], 1, ZZ)
+    g = dmp_normal([[-1], [1, 0]], 1, ZZ)
+
+    q = dmp_normal([[-1], [-1, 0]], 1, ZZ)
+    r = dmp_normal([[2, 0, 0]], 1, ZZ)
+
+    assert dmp_rr_div(f, g, 1, ZZ) == (q, r)
+
+    f = dmp_normal([[1], [], [1, 0, 0]], 1, ZZ)
+    g = dmp_normal([[2], [-2, 0]], 1, ZZ)
+
+    q, r = [[]], f
+
+    assert dmp_rr_div(f, g, 1, ZZ) == (q, r)
+
+
+def test_dup_ff_div():
+    raises(ZeroDivisionError, lambda: dup_ff_div([1, 2, 3], [], QQ))
+
+    f = dup_normal([3, 1, 1, 5], QQ)
+    g = dup_normal([5, -3, 1], QQ)
+
+    q = [QQ(3, 5), QQ(14, 25)]
+    r = [QQ(52, 25), QQ(111, 25)]
+
+    assert dup_ff_div(f, g, QQ) == (q, r)
+
+def test_dup_ff_div_gmpy2():
+    if GROUND_TYPES != 'gmpy2':
+        return
+
+    from gmpy2 import mpq
+    from sympy.polys.domains import GMPYRationalField
+    K = GMPYRationalField()
+
+    f = [mpq(1,3), mpq(3,2)]
+    g = [mpq(2,1)]
+    assert dmp_ff_div(f, g, 0, K) == ([mpq(1,6), mpq(3,4)], [])
+
+    f = [mpq(1,2), mpq(1,3), mpq(1,4), mpq(1,5)]
+    g = [mpq(-1,1), mpq(1,1), mpq(-1,1)]
+    assert dmp_ff_div(f, g, 0, K) == ([mpq(-1,2), mpq(-5,6)], [mpq(7,12), mpq(-19,30)])
+
+def test_dmp_ff_div():
+    raises(ZeroDivisionError, lambda: dmp_ff_div([[1, 2], [3]], [[]], 1, QQ))
+
+    f = dmp_normal([[1], [], [1, 0, 0]], 1, QQ)
+    g = dmp_normal([[1], [-1, 0]], 1, QQ)
+
+    q = [[QQ(1, 1)], [QQ(1, 1), QQ(0, 1)]]
+    r = [[QQ(2, 1), QQ(0, 1), QQ(0, 1)]]
+
+    assert dmp_ff_div(f, g, 1, QQ) == (q, r)
+
+    f = dmp_normal([[1], [], [1, 0, 0]], 1, QQ)
+    g = dmp_normal([[-1], [1, 0]], 1, QQ)
+
+    q = [[QQ(-1, 1)], [QQ(-1, 1), QQ(0, 1)]]
+    r = [[QQ(2, 1), QQ(0, 1), QQ(0, 1)]]
+
+    assert dmp_ff_div(f, g, 1, QQ) == (q, r)
+
+    f = dmp_normal([[1], [], [1, 0, 0]], 1, QQ)
+    g = dmp_normal([[2], [-2, 0]], 1, QQ)
+
+    q = [[QQ(1, 2)], [QQ(1, 2), QQ(0, 1)]]
+    r = [[QQ(2, 1), QQ(0, 1), QQ(0, 1)]]
+
+    assert dmp_ff_div(f, g, 1, QQ) == (q, r)
+
+
+def test_dup_div():
+    f, g, q, r = [5, 4, 3, 2, 1], [1, 2, 3], [5, -6, 0], [20, 1]
+
+    assert dup_div(f, g, ZZ) == (q, r)
+    assert dup_quo(f, g, ZZ) == q
+    assert dup_rem(f, g, ZZ) == r
+
+    raises(ExactQuotientFailed, lambda: dup_exquo(f, g, ZZ))
+
+    f, g, q, r = [5, 4, 3, 2, 1, 0], [1, 2, 0, 0, 9], [5, -6], [15, 2, -44, 54]
+
+    assert dup_div(f, g, ZZ) == (q, r)
+    assert dup_quo(f, g, ZZ) == q
+    assert dup_rem(f, g, ZZ) == r
+
+    raises(ExactQuotientFailed, lambda: dup_exquo(f, g, ZZ))
+
+
+def test_dmp_div():
+    f, g, q, r = [5, 4, 3, 2, 1], [1, 2, 3], [5, -6, 0], [20, 1]
+
+    assert dmp_div(f, g, 0, ZZ) == (q, r)
+    assert dmp_quo(f, g, 0, ZZ) == q
+    assert dmp_rem(f, g, 0, ZZ) == r
+
+    raises(ExactQuotientFailed, lambda: dmp_exquo(f, g, 0, ZZ))
+
+    f, g, q, r = [[[1]]], [[[2]], [1]], [[[]]], [[[1]]]
+
+    assert dmp_div(f, g, 2, ZZ) == (q, r)
+    assert dmp_quo(f, g, 2, ZZ) == q
+    assert dmp_rem(f, g, 2, ZZ) == r
+
+    raises(ExactQuotientFailed, lambda: dmp_exquo(f, g, 2, ZZ))
+
+
+def test_dup_max_norm():
+    assert dup_max_norm([], ZZ) == 0
+    assert dup_max_norm([1], ZZ) == 1
+
+    assert dup_max_norm([1, 4, 2, 3], ZZ) == 4
+
+
+def test_dmp_max_norm():
+    assert dmp_max_norm([[[]]], 2, ZZ) == 0
+    assert dmp_max_norm([[[1]]], 2, ZZ) == 1
+
+    assert dmp_max_norm(f_0, 2, ZZ) == 6
+
+
+def test_dup_l1_norm():
+    assert dup_l1_norm([], ZZ) == 0
+    assert dup_l1_norm([1], ZZ) == 1
+    assert dup_l1_norm([1, 4, 2, 3], ZZ) == 10
+
+
+def test_dmp_l1_norm():
+    assert dmp_l1_norm([[[]]], 2, ZZ) == 0
+    assert dmp_l1_norm([[[1]]], 2, ZZ) == 1
+
+    assert dmp_l1_norm(f_0, 2, ZZ) == 31
+
+
+def test_dup_l2_norm_squared():
+    assert dup_l2_norm_squared([], ZZ) == 0
+    assert dup_l2_norm_squared([1], ZZ) == 1
+    assert dup_l2_norm_squared([1, 4, 2, 3], ZZ) == 30
+
+
+def test_dmp_l2_norm_squared():
+    assert dmp_l2_norm_squared([[[]]], 2, ZZ) == 0
+    assert dmp_l2_norm_squared([[[1]]], 2, ZZ) == 1
+    assert dmp_l2_norm_squared(f_0, 2, ZZ) == 111
+
+
+def test_dup_expand():
+    assert dup_expand((), ZZ) == [1]
+    assert dup_expand(([1, 2, 3], [1, 2], [7, 5, 4, 3]), ZZ) == \
+        dup_mul([1, 2, 3], dup_mul([1, 2], [7, 5, 4, 3], ZZ), ZZ)
+
+
+def test_dmp_expand():
+    assert dmp_expand((), 1, ZZ) == [[1]]
+    assert dmp_expand(([[1], [2], [3]], [[1], [2]], [[7], [5], [4], [3]]), 1, ZZ) == \
+        dmp_mul([[1], [2], [3]], dmp_mul([[1], [2]], [[7], [5], [
+                4], [3]], 1, ZZ), 1, ZZ)
+
+def test_dup_mul_poly():
+    p = Poly(18786186952704.0*x**165 + 9.31746684052255e+31*x**82, x, domain='RR')
+    px = Poly(18786186952704.0*x**166 + 9.31746684052255e+31*x**83, x, domain='RR')
+
+    assert p * x == px
+    assert p.set_domain(QQ) * x == px.set_domain(QQ)
+    assert p.set_domain(CC) * x == px.set_domain(CC)

.venv/lib/python3.13/site-packages/sympy/polys/tests/test_densebasic.py

@@ -0,0 +1,730 @@
+"""Tests for dense recursive polynomials' basic tools. """
+
+from sympy.polys.densebasic import (
+    ninf,
+    dup_LC, dmp_LC,
+    dup_TC, dmp_TC,
+    dmp_ground_LC, dmp_ground_TC,
+    dmp_true_LT,
+    dup_degree, dmp_degree,
+    dmp_degree_in, dmp_degree_list,
+    dup_strip, dmp_strip,
+    dmp_validate,
+    dup_reverse,
+    dup_copy, dmp_copy,
+    dup_normal, dmp_normal,
+    dup_convert, dmp_convert,
+    dup_from_sympy, dmp_from_sympy,
+    dup_nth, dmp_nth, dmp_ground_nth,
+    dmp_zero_p, dmp_zero,
+    dmp_one_p, dmp_one,
+    dmp_ground_p, dmp_ground,
+    dmp_negative_p, dmp_positive_p,
+    dmp_zeros, dmp_grounds,
+    dup_from_dict, dup_from_raw_dict,
+    dup_to_dict, dup_to_raw_dict,
+    dmp_from_dict, dmp_to_dict,
+    dmp_swap, dmp_permute,
+    dmp_nest, dmp_raise,
+    dup_deflate, dmp_deflate,
+    dup_multi_deflate, dmp_multi_deflate,
+    dup_inflate, dmp_inflate,
+    dmp_exclude, dmp_include,
+    dmp_inject, dmp_eject,
+    dup_terms_gcd, dmp_terms_gcd,
+    dmp_list_terms, dmp_apply_pairs,
+    dup_slice,
+    dup_random,
+)
+
+from sympy.polys.specialpolys import f_polys
+from sympy.polys.domains import ZZ, QQ
+from sympy.polys.rings import ring
+
+from sympy.core.singleton import S
+from sympy.testing.pytest import raises
+
+from sympy.core.numbers import oo
+
+f_0, f_1, f_2, f_3, f_4, f_5, f_6 = [ f.to_dense() for f in f_polys() ]
+
+def test_dup_LC():
+    assert dup_LC([], ZZ) == 0
+    assert dup_LC([2, 3, 4, 5], ZZ) == 2
+
+
+def test_dup_TC():
+    assert dup_TC([], ZZ) == 0
+    assert dup_TC([2, 3, 4, 5], ZZ) == 5
+
+
+def test_dmp_LC():
+    assert dmp_LC([[]], ZZ) == []
+    assert dmp_LC([[2, 3, 4], [5]], ZZ) == [2, 3, 4]
+    assert dmp_LC([[[]]], ZZ) == [[]]
+    assert dmp_LC([[[2], [3, 4]], [[5]]], ZZ) == [[2], [3, 4]]
+
+
+def test_dmp_TC():
+    assert dmp_TC([[]], ZZ) == []
+    assert dmp_TC([[2, 3, 4], [5]], ZZ) == [5]
+    assert dmp_TC([[[]]], ZZ) == [[]]
+    assert dmp_TC([[[2], [3, 4]], [[5]]], ZZ) == [[5]]
+
+
+def test_dmp_ground_LC():
+    assert dmp_ground_LC([[]], 1, ZZ) == 0
+    assert dmp_ground_LC([[2, 3, 4], [5]], 1, ZZ) == 2
+    assert dmp_ground_LC([[[]]], 2, ZZ) == 0
+    assert dmp_ground_LC([[[2], [3, 4]], [[5]]], 2, ZZ) == 2
+
+
+def test_dmp_ground_TC():
+    assert dmp_ground_TC([[]], 1, ZZ) == 0
+    assert dmp_ground_TC([[2, 3, 4], [5]], 1, ZZ) == 5
+    assert dmp_ground_TC([[[]]], 2, ZZ) == 0
+    assert dmp_ground_TC([[[2], [3, 4]], [[5]]], 2, ZZ) == 5
+
+
+def test_dmp_true_LT():
+    assert dmp_true_LT([[]], 1, ZZ) == ((0, 0), 0)
+    assert dmp_true_LT([[7]], 1, ZZ) == ((0, 0), 7)
+
+    assert dmp_true_LT([[1, 0]], 1, ZZ) == ((0, 1), 1)
+    assert dmp_true_LT([[1], []], 1, ZZ) == ((1, 0), 1)
+    assert dmp_true_LT([[1, 0], []], 1, ZZ) == ((1, 1), 1)
+
+
+def test_dup_degree():
+    assert ninf == float('-inf')
+    assert dup_degree([]) is ninf
+    assert dup_degree([1]) == 0
+    assert dup_degree([1, 0]) == 1
+    assert dup_degree([1, 0, 0, 0, 1]) == 4
+
+
+def test_dmp_degree():
+    assert dmp_degree([[]], 1) is ninf
+    assert dmp_degree([[[]]], 2) is ninf
+
+    assert dmp_degree([[1]], 1) == 0
+    assert dmp_degree([[2], [1]], 1) == 1
+
+
+def test_dmp_degree_in():
+    assert dmp_degree_in([[[]]], 0, 2) is ninf
+    assert dmp_degree_in([[[]]], 1, 2) is ninf
+    assert dmp_degree_in([[[]]], 2, 2) is ninf
+
+    assert dmp_degree_in([[[1]]], 0, 2) == 0
+    assert dmp_degree_in([[[1]]], 1, 2) == 0
+    assert dmp_degree_in([[[1]]], 2, 2) == 0
+
+    assert dmp_degree_in(f_4, 0, 2) == 9
+    assert dmp_degree_in(f_4, 1, 2) == 12
+    assert dmp_degree_in(f_4, 2, 2) == 8
+
+    assert dmp_degree_in(f_6, 0, 2) == 4
+    assert dmp_degree_in(f_6, 1, 2) == 4
+    assert dmp_degree_in(f_6, 2, 2) == 6
+    assert dmp_degree_in(f_6, 3, 3) == 3
+
+    raises(IndexError, lambda: dmp_degree_in([[1]], -5, 1))
+
+
+def test_dmp_degree_list():
+    assert dmp_degree_list([[[[ ]]]], 3) == (-oo, -oo, -oo, -oo)
+    assert dmp_degree_list([[[[1]]]], 3) == ( 0, 0, 0, 0)
+
+    assert dmp_degree_list(f_0, 2) == (2, 2, 2)
+    assert dmp_degree_list(f_1, 2) == (3, 3, 3)
+    assert dmp_degree_list(f_2, 2) == (5, 3, 3)
+    assert dmp_degree_list(f_3, 2) == (5, 4, 7)
+    assert dmp_degree_list(f_4, 2) == (9, 12, 8)
+    assert dmp_degree_list(f_5, 2) == (3, 3, 3)
+    assert dmp_degree_list(f_6, 3) == (4, 4, 6, 3)
+
+
+def test_dup_strip():
+    assert dup_strip([]) == []
+    assert dup_strip([0]) == []
+    assert dup_strip([0, 0, 0]) == []
+
+    assert dup_strip([1]) == [1]
+    assert dup_strip([0, 1]) == [1]
+    assert dup_strip([0, 0, 0, 1]) == [1]
+
+    assert dup_strip([1, 2, 0]) == [1, 2, 0]
+    assert dup_strip([0, 1, 2, 0]) == [1, 2, 0]
+    assert dup_strip([0, 0, 0, 1, 2, 0]) == [1, 2, 0]
+
+
+def test_dmp_strip():
+    assert dmp_strip([0, 1, 0], 0) == [1, 0]
+
+    assert dmp_strip([[]], 1) == [[]]
+    assert dmp_strip([[], []], 1) == [[]]
+    assert dmp_strip([[], [], []], 1) == [[]]
+
+    assert dmp_strip([[[]]], 2) == [[[]]]
+    assert dmp_strip([[[]], [[]]], 2) == [[[]]]
+    assert dmp_strip([[[]], [[]], [[]]], 2) == [[[]]]
+
+    assert dmp_strip([[[1]]], 2) == [[[1]]]
+    assert dmp_strip([[[]], [[1]]], 2) == [[[1]]]
+    assert dmp_strip([[[]], [[1]], [[]]], 2) == [[[1]], [[]]]
+
+
+def test_dmp_validate():
+    assert dmp_validate([]) == ([], 0)
+    assert dmp_validate([0, 0, 0, 1, 0]) == ([1, 0], 0)
+
+    assert dmp_validate([[[]]]) == ([[[]]], 2)
+    assert dmp_validate([[0], [], [0], [1], [0]]) == ([[1], []], 1)
+
+    raises(ValueError, lambda: dmp_validate([[0], 0, [0], [1], [0]]))
+
+
+def test_dup_reverse():
+    assert dup_reverse([1, 2, 0, 3]) == [3, 0, 2, 1]
+    assert dup_reverse([1, 2, 3, 0]) == [3, 2, 1]
+
+
+def test_dup_copy():
+    f = [ZZ(1), ZZ(0), ZZ(2)]
+    g = dup_copy(f)
+
+    g[0], g[2] = ZZ(7), ZZ(0)
+
+    assert f != g
+
+
+def test_dmp_copy():
+    f = [[ZZ(1)], [ZZ(2), ZZ(0)]]
+    g = dmp_copy(f, 1)
+
+    g[0][0], g[1][1] = ZZ(7), ZZ(1)
+
+    assert f != g
+
+
+def test_dup_normal():
+    assert dup_normal([0, 0, 2, 1, 0, 11, 0], ZZ) == \
+        [ZZ(2), ZZ(1), ZZ(0), ZZ(11), ZZ(0)]
+
+
+def test_dmp_normal():
+    assert dmp_normal([[0], [], [0, 2, 1], [0], [11], []], 1, ZZ) == \
+        [[ZZ(2), ZZ(1)], [], [ZZ(11)], []]
+
+
+def test_dup_convert():
+    K0, K1 = ZZ['x'], ZZ
+
+    f = [K0(1), K0(2), K0(0), K0(3)]
+
+    assert dup_convert(f, K0, K1) == \
+        [ZZ(1), ZZ(2), ZZ(0), ZZ(3)]
+
+
+def test_dmp_convert():
+    K0, K1 = ZZ['x'], ZZ
+
+    f = [[K0(1)], [K0(2)], [], [K0(3)]]
+
+    assert dmp_convert(f, 1, K0, K1) == \
+        [[ZZ(1)], [ZZ(2)], [], [ZZ(3)]]
+
+
+def test_dup_from_sympy():
+    assert dup_from_sympy([S.One, S(2)], ZZ) == \
+        [ZZ(1), ZZ(2)]
+    assert dup_from_sympy([S.Half, S(3)], QQ) == \
+        [QQ(1, 2), QQ(3, 1)]
+
+
+def test_dmp_from_sympy():
+    assert dmp_from_sympy([[S.One, S(2)], [S.Zero]], 1, ZZ) == \
+        [[ZZ(1), ZZ(2)], []]
+    assert dmp_from_sympy([[S.Half, S(2)]], 1, QQ) == \
+        [[QQ(1, 2), QQ(2, 1)]]
+
+
+def test_dup_nth():
+    assert dup_nth([1, 2, 3], 0, ZZ) == 3
+    assert dup_nth([1, 2, 3], 1, ZZ) == 2
+    assert dup_nth([1, 2, 3], 2, ZZ) == 1
+
+    assert dup_nth([1, 2, 3], 9, ZZ) == 0
+
+    raises(IndexError, lambda: dup_nth([3, 4, 5], -1, ZZ))
+
+
+def test_dmp_nth():
+    assert dmp_nth([[1], [2], [3]], 0, 1, ZZ) == [3]
+    assert dmp_nth([[1], [2], [3]], 1, 1, ZZ) == [2]
+    assert dmp_nth([[1], [2], [3]], 2, 1, ZZ) == [1]
+
+    assert dmp_nth([[1], [2], [3]], 9, 1, ZZ) == []
+
+    raises(IndexError, lambda: dmp_nth([[3], [4], [5]], -1, 1, ZZ))
+
+
+def test_dmp_ground_nth():
+    assert dmp_ground_nth([[]], (0, 0), 1, ZZ) == 0
+    assert dmp_ground_nth([[1], [2], [3]], (0, 0), 1, ZZ) == 3
+    assert dmp_ground_nth([[1], [2], [3]], (1, 0), 1, ZZ) == 2
+    assert dmp_ground_nth([[1], [2], [3]], (2, 0), 1, ZZ) == 1
+
+    assert dmp_ground_nth([[1], [2], [3]], (2, 1), 1, ZZ) == 0
+    assert dmp_ground_nth([[1], [2], [3]], (3, 0), 1, ZZ) == 0
+
+    raises(IndexError, lambda: dmp_ground_nth([[3], [4], [5]], (2, -1), 1, ZZ))
+
+
+def test_dmp_zero_p():
+    assert dmp_zero_p([], 0) is True
+    assert dmp_zero_p([[]], 1) is True
+
+    assert dmp_zero_p([[[]]], 2) is True
+    assert dmp_zero_p([[[1]]], 2) is False
+
+
+def test_dmp_zero():
+    assert dmp_zero(0) == []
+    assert dmp_zero(2) == [[[]]]
+
+
+def test_dmp_one_p():
+    assert dmp_one_p([1], 0, ZZ) is True
+    assert dmp_one_p([[1]], 1, ZZ) is True
+    assert dmp_one_p([[[1]]], 2, ZZ) is True
+    assert dmp_one_p([[[12]]], 2, ZZ) is False
+
+
+def test_dmp_one():
+    assert dmp_one(0, ZZ) == [ZZ(1)]
+    assert dmp_one(2, ZZ) == [[[ZZ(1)]]]
+
+
+def test_dmp_ground_p():
+    assert dmp_ground_p([], 0, 0) is True
+    assert dmp_ground_p([[]], 0, 1) is True
+    assert dmp_ground_p([[]], 1, 1) is False
+
+    assert dmp_ground_p([[ZZ(1)]], 1, 1) is True
+    assert dmp_ground_p([[[ZZ(2)]]], 2, 2) is True
+
+    assert dmp_ground_p([[[ZZ(2)]]], 3, 2) is False
+    assert dmp_ground_p([[[ZZ(3)], []]], 3, 2) is False
+
+    assert dmp_ground_p([], None, 0) is True
+    assert dmp_ground_p([[]], None, 1) is True
+
+    assert dmp_ground_p([ZZ(1)], None, 0) is True
+    assert dmp_ground_p([[[ZZ(1)]]], None, 2) is True
+
+    assert dmp_ground_p([[[ZZ(3)], []]], None, 2) is False
+
+
+def test_dmp_ground():
+    assert dmp_ground(ZZ(0), 2) == [[[]]]
+
+    assert dmp_ground(ZZ(7), -1) == ZZ(7)
+    assert dmp_ground(ZZ(7), 0) == [ZZ(7)]
+    assert dmp_ground(ZZ(7), 2) == [[[ZZ(7)]]]
+
+
+def test_dmp_zeros():
+    assert dmp_zeros(4, 0, ZZ) == [[], [], [], []]
+
+    assert dmp_zeros(0, 2, ZZ) == []
+    assert dmp_zeros(1, 2, ZZ) == [[[[]]]]
+    assert dmp_zeros(2, 2, ZZ) == [[[[]]], [[[]]]]
+    assert dmp_zeros(3, 2, ZZ) == [[[[]]], [[[]]], [[[]]]]
+
+    assert dmp_zeros(3, -1, ZZ) == [0, 0, 0]
+
+
+def test_dmp_grounds():
+    assert dmp_grounds(ZZ(7), 0, 2) == []
+
+    assert dmp_grounds(ZZ(7), 1, 2) == [[[[7]]]]
+    assert dmp_grounds(ZZ(7), 2, 2) == [[[[7]]], [[[7]]]]
+    assert dmp_grounds(ZZ(7), 3, 2) == [[[[7]]], [[[7]]], [[[7]]]]
+
+    assert dmp_grounds(ZZ(7), 3, -1) == [7, 7, 7]
+
+
+def test_dmp_negative_p():
+    assert dmp_negative_p([[[]]], 2, ZZ) is False
+    assert dmp_negative_p([[[1], [2]]], 2, ZZ) is False
+    assert dmp_negative_p([[[-1], [2]]], 2, ZZ) is True
+
+
+def test_dmp_positive_p():
+    assert dmp_positive_p([[[]]], 2, ZZ) is False
+    assert dmp_positive_p([[[1], [2]]], 2, ZZ) is True
+    assert dmp_positive_p([[[-1], [2]]], 2, ZZ) is False
+
+
+def test_dup_from_to_dict():
+    assert dup_from_raw_dict({}, ZZ) == []
+    assert dup_from_dict({}, ZZ) == []
+
+    assert dup_to_raw_dict([]) == {}
+    assert dup_to_dict([]) == {}
+
+    assert dup_to_raw_dict([], ZZ, zero=True) == {0: ZZ(0)}
+    assert dup_to_dict([], ZZ, zero=True) == {(0,): ZZ(0)}
+
+    f = [3, 0, 0, 2, 0, 0, 0, 0, 8]
+    g = {8: 3, 5: 2, 0: 8}
+    h = {(8,): 3, (5,): 2, (0,): 8}
+
+    assert dup_from_raw_dict(g, ZZ) == f
+    assert dup_from_dict(h, ZZ) == f
+
+    assert dup_to_raw_dict(f) == g
+    assert dup_to_dict(f) == h
+
+    R, x,y = ring("x,y", ZZ)
+    K = R.to_domain()
+
+    f = [R(3), R(0), R(2), R(0), R(0), R(8)]
+    g = {5: R(3), 3: R(2), 0: R(8)}
+    h = {(5,): R(3), (3,): R(2), (0,): R(8)}
+
+    assert dup_from_raw_dict(g, K) == f
+    assert dup_from_dict(h, K) == f
+
+    assert dup_to_raw_dict(f) == g
+    assert dup_to_dict(f) == h
+
+
+def test_dmp_from_to_dict():
+    assert dmp_from_dict({}, 1, ZZ) == [[]]
+    assert dmp_to_dict([[]], 1) == {}
+
+    assert dmp_to_dict([], 0, ZZ, zero=True) == {(0,): ZZ(0)}
+    assert dmp_to_dict([[]], 1, ZZ, zero=True) == {(0, 0): ZZ(0)}
+
+    f = [[3], [], [], [2], [], [], [], [], [8]]
+    g = {(8, 0): 3, (5, 0): 2, (0, 0): 8}
+
+    assert dmp_from_dict(g, 1, ZZ) == f
+    assert dmp_to_dict(f, 1) == g
+
+
+def test_dmp_swap():
+    f = dmp_normal([[1, 0, 0], [], [1, 0], [], [1]], 1, ZZ)
+    g = dmp_normal([[1, 0, 0, 0, 0], [1, 0, 0], [1]], 1, ZZ)
+
+    assert dmp_swap(f, 1, 1, 1, ZZ) == f
+
+    assert dmp_swap(f, 0, 1, 1, ZZ) == g
+    assert dmp_swap(g, 0, 1, 1, ZZ) == f
+
+    raises(IndexError, lambda: dmp_swap(f, -1, -7, 1, ZZ))
+
+
+def test_dmp_permute():
+    f = dmp_normal([[1, 0, 0], [], [1, 0], [], [1]], 1, ZZ)
+    g = dmp_normal([[1, 0, 0, 0, 0], [1, 0, 0], [1]], 1, ZZ)
+
+    assert dmp_permute(f, [0, 1], 1, ZZ) == f
+    assert dmp_permute(g, [0, 1], 1, ZZ) == g
+
+    assert dmp_permute(f, [1, 0], 1, ZZ) == g
+    assert dmp_permute(g, [1, 0], 1, ZZ) == f
+
+
+def test_dmp_nest():
+    assert dmp_nest(ZZ(1), 2, ZZ) == [[[1]]]
+
+    assert dmp_nest([[1]], 0, ZZ) == [[1]]
+    assert dmp_nest([[1]], 1, ZZ) == [[[1]]]
+    assert dmp_nest([[1]], 2, ZZ) == [[[[1]]]]
+
+
+def test_dmp_raise():
+    assert dmp_raise([], 2, 0, ZZ) == [[[]]]
+    assert dmp_raise([[1]], 0, 1, ZZ) == [[1]]
+
+    assert dmp_raise([[1, 2, 3], [], [2, 3]], 2, 1, ZZ) == \
+        [[[[1]], [[2]], [[3]]], [[[]]], [[[2]], [[3]]]]
+
+
+def test_dup_deflate():
+    assert dup_deflate([], ZZ) == (1, [])
+    assert dup_deflate([2], ZZ) == (1, [2])
+    assert dup_deflate([1, 2, 3], ZZ) == (1, [1, 2, 3])
+    assert dup_deflate([1, 0, 2, 0, 3], ZZ) == (2, [1, 2, 3])
+
+    assert dup_deflate(dup_from_raw_dict({7: 1, 1: 1}, ZZ), ZZ) == \
+        (1, [1, 0, 0, 0, 0, 0, 1, 0])
+    assert dup_deflate(dup_from_raw_dict({7: 1, 0: 1}, ZZ), ZZ) == \
+        (7, [1, 1])
+    assert dup_deflate(dup_from_raw_dict({7: 1, 3: 1}, ZZ), ZZ) == \
+        (1, [1, 0, 0, 0, 1, 0, 0, 0])
+
+    assert dup_deflate(dup_from_raw_dict({7: 1, 4: 1}, ZZ), ZZ) == \
+        (1, [1, 0, 0, 1, 0, 0, 0, 0])
+    assert dup_deflate(dup_from_raw_dict({8: 1, 4: 1}, ZZ), ZZ) == \
+        (4, [1, 1, 0])
+
+    assert dup_deflate(dup_from_raw_dict({8: 1}, ZZ), ZZ) == \
+        (8, [1, 0])
+    assert dup_deflate(dup_from_raw_dict({7: 1}, ZZ), ZZ) == \
+        (7, [1, 0])
+    assert dup_deflate(dup_from_raw_dict({1: 1}, ZZ), ZZ) == \
+        (1, [1, 0])
+
+
+def test_dmp_deflate():
+    assert dmp_deflate([[]], 1, ZZ) == ((1, 1), [[]])
+    assert dmp_deflate([[2]], 1, ZZ) == ((1, 1), [[2]])
+
+    f = [[1, 0, 0], [], [1, 0], [], [1]]
+
+    assert dmp_deflate(f, 1, ZZ) == ((2, 1), [[1, 0, 0], [1, 0], [1]])
+
+
+def test_dup_multi_deflate():
+    assert dup_multi_deflate(([2],), ZZ) == (1, ([2],))
+    assert dup_multi_deflate(([], []), ZZ) == (1, ([], []))
+
+    assert dup_multi_deflate(([1, 2, 3],), ZZ) == (1, ([1, 2, 3],))
+    assert dup_multi_deflate(([1, 0, 2, 0, 3],), ZZ) == (2, ([1, 2, 3],))
+
+    assert dup_multi_deflate(([1, 0, 2, 0, 3], [2, 0, 0]), ZZ) == \
+        (2, ([1, 2, 3], [2, 0]))
+    assert dup_multi_deflate(([1, 0, 2, 0, 3], [2, 1, 0]), ZZ) == \
+        (1, ([1, 0, 2, 0, 3], [2, 1, 0]))
+
+
+def test_dmp_multi_deflate():
+    assert dmp_multi_deflate(([[]],), 1, ZZ) == \
+        ((1, 1), ([[]],))
+    assert dmp_multi_deflate(([[]], [[]]), 1, ZZ) == \
+        ((1, 1), ([[]], [[]]))
+
+    assert dmp_multi_deflate(([[1]], [[]]), 1, ZZ) == \
+        ((1, 1), ([[1]], [[]]))
+    assert dmp_multi_deflate(([[1]], [[2]]), 1, ZZ) == \
+        ((1, 1), ([[1]], [[2]]))
+    assert dmp_multi_deflate(([[1]], [[2, 0]]), 1, ZZ) == \
+        ((1, 1), ([[1]], [[2, 0]]))
+
+    assert dmp_multi_deflate(([[2, 0]], [[2, 0]]), 1, ZZ) == \
+        ((1, 1), ([[2, 0]], [[2, 0]]))
+
+    assert dmp_multi_deflate(
+        ([[2]], [[2, 0, 0]]), 1, ZZ) == ((1, 2), ([[2]], [[2, 0]]))
+    assert dmp_multi_deflate(
+        ([[2, 0, 0]], [[2, 0, 0]]), 1, ZZ) == ((1, 2), ([[2, 0]], [[2, 0]]))
+
+    assert dmp_multi_deflate(([2, 0, 0], [1, 0, 4, 0, 1]), 0, ZZ) == \
+        ((2,), ([2, 0], [1, 4, 1]))
+
+    f = [[1, 0, 0], [], [1, 0], [], [1]]
+    g = [[1, 0, 1, 0], [], [1]]
+
+    assert dmp_multi_deflate((f,), 1, ZZ) == \
+        ((2, 1), ([[1, 0, 0], [1, 0], [1]],))
+
+    assert dmp_multi_deflate((f, g), 1, ZZ) == \
+        ((2, 1), ([[1, 0, 0], [1, 0], [1]],
+                  [[1, 0, 1, 0], [1]]))
+
+
+def test_dup_inflate():
+    assert dup_inflate([], 17, ZZ) == []
+
+    assert dup_inflate([1, 2, 3], 1, ZZ) == [1, 2, 3]
+    assert dup_inflate([1, 2, 3], 2, ZZ) == [1, 0, 2, 0, 3]
+    assert dup_inflate([1, 2, 3], 3, ZZ) == [1, 0, 0, 2, 0, 0, 3]
+    assert dup_inflate([1, 2, 3], 4, ZZ) == [1, 0, 0, 0, 2, 0, 0, 0, 3]
+
+    raises(IndexError, lambda: dup_inflate([1, 2, 3], 0, ZZ))
+
+
+def test_dmp_inflate():
+    assert dmp_inflate([1], (3,), 0, ZZ) == [1]
+
+    assert dmp_inflate([[]], (3, 7), 1, ZZ) == [[]]
+    assert dmp_inflate([[2]], (1, 2), 1, ZZ) == [[2]]
+
+    assert dmp_inflate([[2, 0]], (1, 1), 1, ZZ) == [[2, 0]]
+    assert dmp_inflate([[2, 0]], (1, 2), 1, ZZ) == [[2, 0, 0]]
+    assert dmp_inflate([[2, 0]], (1, 3), 1, ZZ) == [[2, 0, 0, 0]]
+
+    assert dmp_inflate([[1, 0, 0], [1], [1, 0]], (2, 1), 1, ZZ) == \
+        [[1, 0, 0], [], [1], [], [1, 0]]
+
+    raises(IndexError, lambda: dmp_inflate([[]], (-3, 7), 1, ZZ))
+
+
+def test_dmp_exclude():
+    assert dmp_exclude([[[]]], 2, ZZ) == ([], [[[]]], 2)
+    assert dmp_exclude([[[7]]], 2, ZZ) == ([], [[[7]]], 2)
+
+    assert dmp_exclude([1, 2, 3], 0, ZZ) == ([], [1, 2, 3], 0)
+    assert dmp_exclude([[1], [2, 3]], 1, ZZ) == ([], [[1], [2, 3]], 1)
+
+    assert dmp_exclude([[1, 2, 3]], 1, ZZ) == ([0], [1, 2, 3], 0)
+    assert dmp_exclude([[1], [2], [3]], 1, ZZ) == ([1], [1, 2, 3], 0)
+
+    assert dmp_exclude([[[1, 2, 3]]], 2, ZZ) == ([0, 1], [1, 2, 3], 0)
+    assert dmp_exclude([[[1]], [[2]], [[3]]], 2, ZZ) == ([1, 2], [1, 2, 3], 0)
+
+
+def test_dmp_include():
+    assert dmp_include([1, 2, 3], [], 0, ZZ) == [1, 2, 3]
+
+    assert dmp_include([1, 2, 3], [0], 0, ZZ) == [[1, 2, 3]]
+    assert dmp_include([1, 2, 3], [1], 0, ZZ) == [[1], [2], [3]]
+
+    assert dmp_include([1, 2, 3], [0, 1], 0, ZZ) == [[[1, 2, 3]]]
+    assert dmp_include([1, 2, 3], [1, 2], 0, ZZ) == [[[1]], [[2]], [[3]]]
+
+
+def test_dmp_inject():
+    R, x,y = ring("x,y", ZZ)
+    K = R.to_domain()
+
+    assert dmp_inject([], 0, K) == ([[[]]], 2)
+    assert dmp_inject([[]], 1, K) == ([[[[]]]], 3)
+
+    assert dmp_inject([R(1)], 0, K) == ([[[1]]], 2)
+    assert dmp_inject([[R(1)]], 1, K) == ([[[[1]]]], 3)
+
+    assert dmp_inject([R(1), 2*x + 3*y + 4], 0, K) == ([[[1]], [[2], [3, 4]]], 2)
+
+    f = [3*x**2 + 7*x*y + 5*y**2, 2*x, R(0), x*y**2 + 11]
+    g = [[[3], [7, 0], [5, 0, 0]], [[2], []], [[]], [[1, 0, 0], [11]]]
+
+    assert dmp_inject(f, 0, K) == (g, 2)
+
+
+def test_dmp_eject():
+    R, x,y = ring("x,y", ZZ)
+    K = R.to_domain()
+
+    assert dmp_eject([[[]]], 2, K) == []
+    assert dmp_eject([[[[]]]], 3, K) == [[]]
+
+    assert dmp_eject([[[1]]], 2, K) == [R(1)]
+    assert dmp_eject([[[[1]]]], 3, K) == [[R(1)]]
+
+    assert dmp_eject([[[1]], [[2], [3, 4]]], 2, K) == [R(1), 2*x + 3*y + 4]
+
+    f = [3*x**2 + 7*x*y + 5*y**2, 2*x, R(0), x*y**2 + 11]
+    g = [[[3], [7, 0], [5, 0, 0]], [[2], []], [[]], [[1, 0, 0], [11]]]
+
+    assert dmp_eject(g, 2, K) == f
+
+
+def test_dup_terms_gcd():
+    assert dup_terms_gcd([], ZZ) == (0, [])
+    assert dup_terms_gcd([1, 0, 1], ZZ) == (0, [1, 0, 1])
+    assert dup_terms_gcd([1, 0, 1, 0], ZZ) == (1, [1, 0, 1])
+
+
+def test_dmp_terms_gcd():
+    assert dmp_terms_gcd([[]], 1, ZZ) == ((0, 0), [[]])
+
+    assert dmp_terms_gcd([1, 0, 1, 0], 0, ZZ) == ((1,), [1, 0, 1])
+    assert dmp_terms_gcd([[1], [], [1], []], 1, ZZ) == ((1, 0), [[1], [], [1]])
+
+    assert dmp_terms_gcd(
+        [[1, 0], [], [1]], 1, ZZ) == ((0, 0), [[1, 0], [], [1]])
+    assert dmp_terms_gcd(
+        [[1, 0], [1, 0, 0], [], []], 1, ZZ) == ((2, 1), [[1], [1, 0]])
+
+
+def test_dmp_list_terms():
+    assert dmp_list_terms([[[]]], 2, ZZ) == [((0, 0, 0), 0)]
+    assert dmp_list_terms([[[1]]], 2, ZZ) == [((0, 0, 0), 1)]
+
+    assert dmp_list_terms([1, 2, 4, 3, 5], 0, ZZ) == \
+        [((4,), 1), ((3,), 2), ((2,), 4), ((1,), 3), ((0,), 5)]
+
+    assert dmp_list_terms([[1], [2, 4], [3, 5, 0]], 1, ZZ) == \
+        [((2, 0), 1), ((1, 1), 2), ((1, 0), 4), ((0, 2), 3), ((0, 1), 5)]
+
+    f = [[2, 0, 0, 0], [1, 0, 0], []]
+
+    assert dmp_list_terms(f, 1, ZZ, order='lex') == [((2, 3), 2), ((1, 2), 1)]
+    assert dmp_list_terms(
+        f, 1, ZZ, order='grlex') == [((2, 3), 2), ((1, 2), 1)]
+
+    f = [[2, 0, 0, 0], [1, 0, 0, 0, 0, 0], []]
+
+    assert dmp_list_terms(f, 1, ZZ, order='lex') == [((2, 3), 2), ((1, 5), 1)]
+    assert dmp_list_terms(
+        f, 1, ZZ, order='grlex') == [((1, 5), 1), ((2, 3), 2)]
+
+
+def test_dmp_apply_pairs():
+    h = lambda a, b: a*b
+
+    assert dmp_apply_pairs([1, 2, 3], [4, 5, 6], h, [], 0, ZZ) == [4, 10, 18]
+
+    assert dmp_apply_pairs([2, 3], [4, 5, 6], h, [], 0, ZZ) == [10, 18]
+    assert dmp_apply_pairs([1, 2, 3], [5, 6], h, [], 0, ZZ) == [10, 18]
+
+    assert dmp_apply_pairs(
+        [[1, 2], [3]], [[4, 5], [6]], h, [], 1, ZZ) == [[4, 10], [18]]
+
+    assert dmp_apply_pairs(
+        [[1, 2], [3]], [[4], [5, 6]], h, [], 1, ZZ) == [[8], [18]]
+    assert dmp_apply_pairs(
+        [[1], [2, 3]], [[4, 5], [6]], h, [], 1, ZZ) == [[5], [18]]
+
+
+def test_dup_slice():
+    f = [1, 2, 3, 4]
+
+    assert dup_slice(f, 0, 0, ZZ) == []
+    assert dup_slice(f, 0, 1, ZZ) == [4]
+    assert dup_slice(f, 0, 2, ZZ) == [3, 4]
+    assert dup_slice(f, 0, 3, ZZ) == [2, 3, 4]
+    assert dup_slice(f, 0, 4, ZZ) == [1, 2, 3, 4]
+
+    assert dup_slice(f, 0, 4, ZZ) == f
+    assert dup_slice(f, 0, 9, ZZ) == f
+
+    assert dup_slice(f, 1, 0, ZZ) == []
+    assert dup_slice(f, 1, 1, ZZ) == []
+    assert dup_slice(f, 1, 2, ZZ) == [3, 0]
+    assert dup_slice(f, 1, 3, ZZ) == [2, 3, 0]
+    assert dup_slice(f, 1, 4, ZZ) == [1, 2, 3, 0]
+
+    assert dup_slice([1, 2], 0, 3, ZZ) == [1, 2]
+
+    g = [1, 0, 0, 2]
+
+    assert dup_slice(g, 0, 3, ZZ) == [2]
+
+
+def test_dup_random():
+    f = dup_random(0, -10, 10, ZZ)
+
+    assert dup_degree(f) == 0
+    assert all(-10 <= c <= 10 for c in f)
+
+    f = dup_random(1, -20, 20, ZZ)
+
+    assert dup_degree(f) == 1
+    assert all(-20 <= c <= 20 for c in f)
+
+    f = dup_random(2, -30, 30, ZZ)
+
+    assert dup_degree(f) == 2
+    assert all(-30 <= c <= 30 for c in f)
+
+    f = dup_random(3, -40, 40, ZZ)
+
+    assert dup_degree(f) == 3
+    assert all(-40 <= c <= 40 for c in f)

.venv/lib/python3.13/site-packages/sympy/polys/tests/test_densetools.py

@@ -0,0 +1,714 @@
+"""Tests for dense recursive polynomials' tools. """
+
+from sympy.polys.densebasic import (
+    dup_normal, dmp_normal,
+    dup_from_raw_dict,
+    dmp_convert, dmp_swap,
+)
+
+from sympy.polys.densearith import dmp_mul_ground
+
+from sympy.polys.densetools import (
+    dup_clear_denoms, dmp_clear_denoms,
+    dup_integrate, dmp_integrate, dmp_integrate_in,
+    dup_diff, dmp_diff, dmp_diff_in,
+    dup_eval, dmp_eval, dmp_eval_in,
+    dmp_eval_tail, dmp_diff_eval_in,
+    dup_trunc, dmp_trunc, dmp_ground_trunc,
+    dup_monic, dmp_ground_monic,
+    dup_content, dmp_ground_content,
+    dup_primitive, dmp_ground_primitive,
+    dup_extract, dmp_ground_extract,
+    dup_real_imag,
+    dup_mirror, dup_scale, dup_shift, dmp_shift,
+    dup_transform,
+    dup_compose, dmp_compose,
+    dup_decompose,
+    dmp_lift,
+    dup_sign_variations,
+    dup_revert, dmp_revert,
+)
+from sympy.polys.polyclasses import ANP
+
+from sympy.polys.polyerrors import (
+    MultivariatePolynomialError,
+    ExactQuotientFailed,
+    NotReversible,
+    DomainError,
+)
+
+from sympy.polys.specialpolys import f_polys
+
+from sympy.polys.domains import FF, ZZ, QQ, ZZ_I, QQ_I, EX, RR
+from sympy.polys.rings import ring
+
+from sympy.core.numbers import I
+from sympy.core.singleton import S
+from sympy.functions.elementary.trigonometric import sin
+
+from sympy.abc import x
+from sympy.testing.pytest import raises
+
+f_0, f_1, f_2, f_3, f_4, f_5, f_6 = [ f.to_dense() for f in f_polys() ]
+
+def test_dup_integrate():
+    assert dup_integrate([], 1, QQ) == []
+    assert dup_integrate([], 2, QQ) == []
+
+    assert dup_integrate([QQ(1)], 1, QQ) == [QQ(1), QQ(0)]
+    assert dup_integrate([QQ(1)], 2, QQ) == [QQ(1, 2), QQ(0), QQ(0)]
+
+    assert dup_integrate([QQ(1), QQ(2), QQ(3)], 0, QQ) == \
+        [QQ(1), QQ(2), QQ(3)]
+    assert dup_integrate([QQ(1), QQ(2), QQ(3)], 1, QQ) == \
+        [QQ(1, 3), QQ(1), QQ(3), QQ(0)]
+    assert dup_integrate([QQ(1), QQ(2), QQ(3)], 2, QQ) == \
+        [QQ(1, 12), QQ(1, 3), QQ(3, 2), QQ(0), QQ(0)]
+    assert dup_integrate([QQ(1), QQ(2), QQ(3)], 3, QQ) == \
+        [QQ(1, 60), QQ(1, 12), QQ(1, 2), QQ(0), QQ(0), QQ(0)]
+
+    assert dup_integrate(dup_from_raw_dict({29: QQ(17)}, QQ), 3, QQ) == \
+        dup_from_raw_dict({32: QQ(17, 29760)}, QQ)
+
+    assert dup_integrate(dup_from_raw_dict({29: QQ(17), 5: QQ(1, 2)}, QQ), 3, QQ) == \
+        dup_from_raw_dict({32: QQ(17, 29760), 8: QQ(1, 672)}, QQ)
+
+
+def test_dmp_integrate():
+    assert dmp_integrate([QQ(1)], 2, 0, QQ) == [QQ(1, 2), QQ(0), QQ(0)]
+
+    assert dmp_integrate([[[]]], 1, 2, QQ) == [[[]]]
+    assert dmp_integrate([[[]]], 2, 2, QQ) == [[[]]]
+
+    assert dmp_integrate([[[QQ(1)]]], 1, 2, QQ) == [[[QQ(1)]], [[]]]
+    assert dmp_integrate([[[QQ(1)]]], 2, 2, QQ) == [[[QQ(1, 2)]], [[]], [[]]]
+
+    assert dmp_integrate([[QQ(1)], [QQ(2)], [QQ(3)]], 0, 1, QQ) == \
+        [[QQ(1)], [QQ(2)], [QQ(3)]]
+    assert dmp_integrate([[QQ(1)], [QQ(2)], [QQ(3)]], 1, 1, QQ) == \
+        [[QQ(1, 3)], [QQ(1)], [QQ(3)], []]
+    assert dmp_integrate([[QQ(1)], [QQ(2)], [QQ(3)]], 2, 1, QQ) == \
+        [[QQ(1, 12)], [QQ(1, 3)], [QQ(3, 2)], [], []]
+    assert dmp_integrate([[QQ(1)], [QQ(2)], [QQ(3)]], 3, 1, QQ) == \
+        [[QQ(1, 60)], [QQ(1, 12)], [QQ(1, 2)], [], [], []]
+
+
+def test_dmp_integrate_in():
+    f = dmp_convert(f_6, 3, ZZ, QQ)
+
+    assert dmp_integrate_in(f, 2, 1, 3, QQ) == \
+        dmp_swap(
+            dmp_integrate(dmp_swap(f, 0, 1, 3, QQ), 2, 3, QQ), 0, 1, 3, QQ)
+    assert dmp_integrate_in(f, 3, 1, 3, QQ) == \
+        dmp_swap(
+            dmp_integrate(dmp_swap(f, 0, 1, 3, QQ), 3, 3, QQ), 0, 1, 3, QQ)
+    assert dmp_integrate_in(f, 2, 2, 3, QQ) == \
+        dmp_swap(
+            dmp_integrate(dmp_swap(f, 0, 2, 3, QQ), 2, 3, QQ), 0, 2, 3, QQ)
+    assert dmp_integrate_in(f, 3, 2, 3, QQ) == \
+        dmp_swap(
+            dmp_integrate(dmp_swap(f, 0, 2, 3, QQ), 3, 3, QQ), 0, 2, 3, QQ)
+
+    raises(IndexError, lambda: dmp_integrate_in(f, 1, -1, 3, QQ))
+    raises(IndexError, lambda: dmp_integrate_in(f, 1, 4, 3, QQ))
+
+
+def test_dup_diff():
+    assert dup_diff([], 1, ZZ) == []
+    assert dup_diff([7], 1, ZZ) == []
+    assert dup_diff([2, 7], 1, ZZ) == [2]
+    assert dup_diff([1, 2, 1], 1, ZZ) == [2, 2]
+    assert dup_diff([1, 2, 3, 4], 1, ZZ) == [3, 4, 3]
+    assert dup_diff([1, -1, 0, 0, 2], 1, ZZ) == [4, -3, 0, 0]
+
+    f = dup_normal([17, 34, 56, -345, 23, 76, 0, 0, 12, 3, 7], ZZ)
+
+    assert dup_diff(f, 0, ZZ) == f
+    assert dup_diff(f, 1, ZZ) == [170, 306, 448, -2415, 138, 380, 0, 0, 24, 3]
+    assert dup_diff(f, 2, ZZ) == dup_diff(dup_diff(f, 1, ZZ), 1, ZZ)
+    assert dup_diff(
+        f, 3, ZZ) == dup_diff(dup_diff(dup_diff(f, 1, ZZ), 1, ZZ), 1, ZZ)
+
+    K = FF(3)
+    f = dup_normal([17, 34, 56, -345, 23, 76, 0, 0, 12, 3, 7], K)
+
+    assert dup_diff(f, 1, K) == dup_normal([2, 0, 1, 0, 0, 2, 0, 0, 0, 0], K)
+    assert dup_diff(f, 2, K) == dup_normal([1, 0, 0, 2, 0, 0, 0], K)
+    assert dup_diff(f, 3, K) == dup_normal([], K)
+
+    assert dup_diff(f, 0, K) == f
+    assert dup_diff(f, 2, K) == dup_diff(dup_diff(f, 1, K), 1, K)
+    assert dup_diff(
+        f, 3, K) == dup_diff(dup_diff(dup_diff(f, 1, K), 1, K), 1, K)
+
+
+def test_dmp_diff():
+    assert dmp_diff([], 1, 0, ZZ) == []
+    assert dmp_diff([[]], 1, 1, ZZ) == [[]]
+    assert dmp_diff([[[]]], 1, 2, ZZ) == [[[]]]
+
+    assert dmp_diff([[[1], [2]]], 1, 2, ZZ) == [[[]]]
+
+    assert dmp_diff([[[1]], [[]]], 1, 2, ZZ) == [[[1]]]
+    assert dmp_diff([[[3]], [[1]], [[]]], 1, 2, ZZ) == [[[6]], [[1]]]
+
+    assert dmp_diff([1, -1, 0, 0, 2], 1, 0, ZZ) == \
+        dup_diff([1, -1, 0, 0, 2], 1, ZZ)
+
+    assert dmp_diff(f_6, 0, 3, ZZ) == f_6
+    assert dmp_diff(f_6, 1, 3, ZZ) == [[[[8460]], [[]]],
+                       [[[135, 0, 0], [], [], [-135, 0, 0]]],
+                       [[[]]],
+                       [[[-423]], [[-47]], [[]], [[141], [], [94, 0], []], [[]]]]
+    assert dmp_diff(
+        f_6, 2, 3, ZZ) == dmp_diff(dmp_diff(f_6, 1, 3, ZZ), 1, 3, ZZ)
+    assert dmp_diff(f_6, 3, 3, ZZ) == dmp_diff(
+        dmp_diff(dmp_diff(f_6, 1, 3, ZZ), 1, 3, ZZ), 1, 3, ZZ)
+
+    K = FF(23)
+    F_6 = dmp_normal(f_6, 3, K)
+
+    assert dmp_diff(F_6, 0, 3, K) == F_6
+    assert dmp_diff(F_6, 1, 3, K) == dmp_diff(F_6, 1, 3, K)
+    assert dmp_diff(F_6, 2, 3, K) == dmp_diff(dmp_diff(F_6, 1, 3, K), 1, 3, K)
+    assert dmp_diff(F_6, 3, 3, K) == dmp_diff(
+        dmp_diff(dmp_diff(F_6, 1, 3, K), 1, 3, K), 1, 3, K)
+
+
+def test_dmp_diff_in():
+    assert dmp_diff_in(f_6, 2, 1, 3, ZZ) == \
+        dmp_swap(dmp_diff(dmp_swap(f_6, 0, 1, 3, ZZ), 2, 3, ZZ), 0, 1, 3, ZZ)
+    assert dmp_diff_in(f_6, 3, 1, 3, ZZ) == \
+        dmp_swap(dmp_diff(dmp_swap(f_6, 0, 1, 3, ZZ), 3, 3, ZZ), 0, 1, 3, ZZ)
+    assert dmp_diff_in(f_6, 2, 2, 3, ZZ) == \
+        dmp_swap(dmp_diff(dmp_swap(f_6, 0, 2, 3, ZZ), 2, 3, ZZ), 0, 2, 3, ZZ)
+    assert dmp_diff_in(f_6, 3, 2, 3, ZZ) == \
+        dmp_swap(dmp_diff(dmp_swap(f_6, 0, 2, 3, ZZ), 3, 3, ZZ), 0, 2, 3, ZZ)
+
+    raises(IndexError, lambda: dmp_diff_in(f_6, 1, -1, 3, ZZ))
+    raises(IndexError, lambda: dmp_diff_in(f_6, 1, 4, 3, ZZ))
+
+def test_dup_eval():
+    assert dup_eval([], 7, ZZ) == 0
+    assert dup_eval([1, 2], 0, ZZ) == 2
+    assert dup_eval([1, 2, 3], 7, ZZ) == 66
+
+
+def test_dmp_eval():
+    assert dmp_eval([], 3, 0, ZZ) == 0
+
+    assert dmp_eval([[]], 3, 1, ZZ) == []
+    assert dmp_eval([[[]]], 3, 2, ZZ) == [[]]
+
+    assert dmp_eval([[1, 2]], 0, 1, ZZ) == [1, 2]
+
+    assert dmp_eval([[[1]]], 3, 2, ZZ) == [[1]]
+    assert dmp_eval([[[1, 2]]], 3, 2, ZZ) == [[1, 2]]
+
+    assert dmp_eval([[3, 2], [1, 2]], 3, 1, ZZ) == [10, 8]
+    assert dmp_eval([[[3, 2]], [[1, 2]]], 3, 2, ZZ) == [[10, 8]]
+
+
+def test_dmp_eval_in():
+    assert dmp_eval_in(
+        f_6, -2, 1, 3, ZZ) == dmp_eval(dmp_swap(f_6, 0, 1, 3, ZZ), -2, 3, ZZ)
+    assert dmp_eval_in(
+        f_6, 7, 1, 3, ZZ) == dmp_eval(dmp_swap(f_6, 0, 1, 3, ZZ), 7, 3, ZZ)
+    assert dmp_eval_in(f_6, -2, 2, 3, ZZ) == dmp_swap(
+        dmp_eval(dmp_swap(f_6, 0, 2, 3, ZZ), -2, 3, ZZ), 0, 1, 2, ZZ)
+    assert dmp_eval_in(f_6, 7, 2, 3, ZZ) == dmp_swap(
+        dmp_eval(dmp_swap(f_6, 0, 2, 3, ZZ), 7, 3, ZZ), 0, 1, 2, ZZ)
+
+    f = [[[int(45)]], [[]], [[]], [[int(-9)], [-1], [], [int(3), int(0), int(10), int(0)]]]
+
+    assert dmp_eval_in(f, -2, 2, 2, ZZ) == \
+        [[45], [], [], [-9, -1, 0, -44]]
+
+    raises(IndexError, lambda: dmp_eval_in(f_6, ZZ(1), -1, 3, ZZ))
+    raises(IndexError, lambda: dmp_eval_in(f_6, ZZ(1), 4, 3, ZZ))
+
+
+def test_dmp_eval_tail():
+    assert dmp_eval_tail([[]], [1], 1, ZZ) == []
+    assert dmp_eval_tail([[[]]], [1], 2, ZZ) == [[]]
+    assert dmp_eval_tail([[[]]], [1, 2], 2, ZZ) == []
+
+    assert dmp_eval_tail(f_0, [], 2, ZZ) == f_0
+
+    assert dmp_eval_tail(f_0, [1, -17, 8], 2, ZZ) == 84496
+    assert dmp_eval_tail(f_0, [-17, 8], 2, ZZ) == [-1409, 3, 85902]
+    assert dmp_eval_tail(f_0, [8], 2, ZZ) == [[83, 2], [3], [302, 81, 1]]
+
+    assert dmp_eval_tail(f_1, [-17, 8], 2, ZZ) == [-136, 15699, 9166, -27144]
+
+    assert dmp_eval_tail(
+        f_2, [-12, 3], 2, ZZ) == [-1377, 0, -702, -1224, 0, -624]
+    assert dmp_eval_tail(
+        f_3, [-12, 3], 2, ZZ) == [144, 82, -5181, -28872, -14868, -540]
+
+    assert dmp_eval_tail(
+        f_4, [25, -1], 2, ZZ) == [152587890625, 9765625, -59605407714843750,
+        -3839159765625, -1562475, 9536712644531250, 610349546750, -4, 24414375000, 1562520]
+    assert dmp_eval_tail(f_5, [25, -1], 2, ZZ) == [-1, -78, -2028, -17576]
+
+    assert dmp_eval_tail(f_6, [0, 2, 4], 3, ZZ) == [5040, 0, 0, 4480]
+
+
+def test_dmp_diff_eval_in():
+    assert dmp_diff_eval_in(f_6, 2, 7, 1, 3, ZZ) == \
+        dmp_eval(dmp_diff(dmp_swap(f_6, 0, 1, 3, ZZ), 2, 3, ZZ), 7, 3, ZZ)
+
+    assert dmp_diff_eval_in(f_6, 2, 7, 0, 3, ZZ) == \
+        dmp_eval(dmp_diff(f_6, 2, 3, ZZ), 7, 3, ZZ)
+
+    raises(IndexError, lambda: dmp_diff_eval_in(f_6, 1, ZZ(1), 4, 3, ZZ))
+
+
+def test_dup_revert():
+    f = [-QQ(1, 720), QQ(0), QQ(1, 24), QQ(0), -QQ(1, 2), QQ(0), QQ(1)]
+    g = [QQ(61, 720), QQ(0), QQ(5, 24), QQ(0), QQ(1, 2), QQ(0), QQ(1)]
+
+    assert dup_revert(f, 8, QQ) == g
+
+    raises(NotReversible, lambda: dup_revert([QQ(1), QQ(0)], 3, QQ))
+
+
+def test_dmp_revert():
+    f = [-QQ(1, 720), QQ(0), QQ(1, 24), QQ(0), -QQ(1, 2), QQ(0), QQ(1)]
+    g = [QQ(61, 720), QQ(0), QQ(5, 24), QQ(0), QQ(1, 2), QQ(0), QQ(1)]
+
+    assert dmp_revert(f, 8, 0, QQ) == g
+
+    raises(MultivariatePolynomialError, lambda: dmp_revert([[1]], 2, 1, QQ))
+
+
+def test_dup_trunc():
+    assert dup_trunc([1, 2, 3, 4, 5, 6], ZZ(3), ZZ) == [1, -1, 0, 1, -1, 0]
+    assert dup_trunc([6, 5, 4, 3, 2, 1], ZZ(3), ZZ) == [-1, 1, 0, -1, 1]
+
+    R = ZZ_I
+    assert dup_trunc([R(3), R(4), R(5)], R(3), R) == [R(1), R(-1)]
+
+    K = FF(5)
+    assert dup_trunc([K(3), K(4), K(5)], K(3), K) == [K(1), K(0)]
+
+
+def test_dmp_trunc():
+    assert dmp_trunc([[]], [1, 2], 2, ZZ) == [[]]
+    assert dmp_trunc([[1, 2], [1, 4, 1], [1]], [1, 2], 1, ZZ) == [[-3], [1]]
+
+
+def test_dmp_ground_trunc():
+    assert dmp_ground_trunc(f_0, ZZ(3), 2, ZZ) == \
+        dmp_normal(
+            [[[1, -1, 0], [-1]], [[]], [[1, -1, 0], [1, -1, 1], [1]]], 2, ZZ)
+
+
+def test_dup_monic():
+    assert dup_monic([3, 6, 9], ZZ) == [1, 2, 3]
+
+    raises(ExactQuotientFailed, lambda: dup_monic([3, 4, 5], ZZ))
+
+    assert dup_monic([], QQ) == []
+    assert dup_monic([QQ(1)], QQ) == [QQ(1)]
+    assert dup_monic([QQ(7), QQ(1), QQ(21)], QQ) == [QQ(1), QQ(1, 7), QQ(3)]
+
+
+def test_dmp_ground_monic():
+    assert dmp_ground_monic([3, 6, 9], 0, ZZ) == [1, 2, 3]
+
+    assert dmp_ground_monic([[3], [6], [9]], 1, ZZ) == [[1], [2], [3]]
+
+    raises(
+        ExactQuotientFailed, lambda: dmp_ground_monic([[3], [4], [5]], 1, ZZ))
+
+    assert dmp_ground_monic([[]], 1, QQ) == [[]]
+    assert dmp_ground_monic([[QQ(1)]], 1, QQ) == [[QQ(1)]]
+    assert dmp_ground_monic(
+        [[QQ(7)], [QQ(1)], [QQ(21)]], 1, QQ) == [[QQ(1)], [QQ(1, 7)], [QQ(3)]]
+
+
+def test_dup_content():
+    assert dup_content([], ZZ) == ZZ(0)
+    assert dup_content([1], ZZ) == ZZ(1)
+    assert dup_content([-1], ZZ) == ZZ(1)
+    assert dup_content([1, 1], ZZ) == ZZ(1)
+    assert dup_content([2, 2], ZZ) == ZZ(2)
+    assert dup_content([1, 2, 1], ZZ) == ZZ(1)
+    assert dup_content([2, 4, 2], ZZ) == ZZ(2)
+
+    assert dup_content([QQ(2, 3), QQ(4, 9)], QQ) == QQ(2, 9)
+    assert dup_content([QQ(2, 3), QQ(4, 5)], QQ) == QQ(2, 15)
+
+
+def test_dmp_ground_content():
+    assert dmp_ground_content([[]], 1, ZZ) == ZZ(0)
+    assert dmp_ground_content([[]], 1, QQ) == QQ(0)
+    assert dmp_ground_content([[1]], 1, ZZ) == ZZ(1)
+    assert dmp_ground_content([[-1]], 1, ZZ) == ZZ(1)
+    assert dmp_ground_content([[1], [1]], 1, ZZ) == ZZ(1)
+    assert dmp_ground_content([[2], [2]], 1, ZZ) == ZZ(2)
+    assert dmp_ground_content([[1], [2], [1]], 1, ZZ) == ZZ(1)
+    assert dmp_ground_content([[2], [4], [2]], 1, ZZ) == ZZ(2)
+
+    assert dmp_ground_content([[QQ(2, 3)], [QQ(4, 9)]], 1, QQ) == QQ(2, 9)
+    assert dmp_ground_content([[QQ(2, 3)], [QQ(4, 5)]], 1, QQ) == QQ(2, 15)
+
+    assert dmp_ground_content(f_0, 2, ZZ) == ZZ(1)
+    assert dmp_ground_content(
+        dmp_mul_ground(f_0, ZZ(2), 2, ZZ), 2, ZZ) == ZZ(2)
+
+    assert dmp_ground_content(f_1, 2, ZZ) == ZZ(1)
+    assert dmp_ground_content(
+        dmp_mul_ground(f_1, ZZ(3), 2, ZZ), 2, ZZ) == ZZ(3)
+
+    assert dmp_ground_content(f_2, 2, ZZ) == ZZ(1)
+    assert dmp_ground_content(
+        dmp_mul_ground(f_2, ZZ(4), 2, ZZ), 2, ZZ) == ZZ(4)
+
+    assert dmp_ground_content(f_3, 2, ZZ) == ZZ(1)
+    assert dmp_ground_content(
+        dmp_mul_ground(f_3, ZZ(5), 2, ZZ), 2, ZZ) == ZZ(5)
+
+    assert dmp_ground_content(f_4, 2, ZZ) == ZZ(1)
+    assert dmp_ground_content(
+        dmp_mul_ground(f_4, ZZ(6), 2, ZZ), 2, ZZ) == ZZ(6)
+
+    assert dmp_ground_content(f_5, 2, ZZ) == ZZ(1)
+    assert dmp_ground_content(
+        dmp_mul_ground(f_5, ZZ(7), 2, ZZ), 2, ZZ) == ZZ(7)
+
+    assert dmp_ground_content(f_6, 3, ZZ) == ZZ(1)
+    assert dmp_ground_content(
+        dmp_mul_ground(f_6, ZZ(8), 3, ZZ), 3, ZZ) == ZZ(8)
+
+
+def test_dup_primitive():
+    assert dup_primitive([], ZZ) == (ZZ(0), [])
+    assert dup_primitive([ZZ(1)], ZZ) == (ZZ(1), [ZZ(1)])
+    assert dup_primitive([ZZ(1), ZZ(1)], ZZ) == (ZZ(1), [ZZ(1), ZZ(1)])
+    assert dup_primitive([ZZ(2), ZZ(2)], ZZ) == (ZZ(2), [ZZ(1), ZZ(1)])
+    assert dup_primitive(
+        [ZZ(1), ZZ(2), ZZ(1)], ZZ) == (ZZ(1), [ZZ(1), ZZ(2), ZZ(1)])
+    assert dup_primitive(
+        [ZZ(2), ZZ(4), ZZ(2)], ZZ) == (ZZ(2), [ZZ(1), ZZ(2), ZZ(1)])
+
+    assert dup_primitive([], QQ) == (QQ(0), [])
+    assert dup_primitive([QQ(1)], QQ) == (QQ(1), [QQ(1)])
+    assert dup_primitive([QQ(1), QQ(1)], QQ) == (QQ(1), [QQ(1), QQ(1)])
+    assert dup_primitive([QQ(2), QQ(2)], QQ) == (QQ(2), [QQ(1), QQ(1)])
+    assert dup_primitive(
+        [QQ(1), QQ(2), QQ(1)], QQ) == (QQ(1), [QQ(1), QQ(2), QQ(1)])
+    assert dup_primitive(
+        [QQ(2), QQ(4), QQ(2)], QQ) == (QQ(2), [QQ(1), QQ(2), QQ(1)])
+
+    assert dup_primitive(
+        [QQ(2, 3), QQ(4, 9)], QQ) == (QQ(2, 9), [QQ(3), QQ(2)])
+    assert dup_primitive(
+        [QQ(2, 3), QQ(4, 5)], QQ) == (QQ(2, 15), [QQ(5), QQ(6)])
+
+
+def test_dmp_ground_primitive():
+    assert dmp_ground_primitive([ZZ(1)], 0, ZZ) == (ZZ(1), [ZZ(1)])
+
+    assert dmp_ground_primitive([[]], 1, ZZ) == (ZZ(0), [[]])
+
+    assert dmp_ground_primitive(f_0, 2, ZZ) == (ZZ(1), f_0)
+    assert dmp_ground_primitive(
+        dmp_mul_ground(f_0, ZZ(2), 2, ZZ), 2, ZZ) == (ZZ(2), f_0)
+
+    assert dmp_ground_primitive(f_1, 2, ZZ) == (ZZ(1), f_1)
+    assert dmp_ground_primitive(
+        dmp_mul_ground(f_1, ZZ(3), 2, ZZ), 2, ZZ) == (ZZ(3), f_1)
+
+    assert dmp_ground_primitive(f_2, 2, ZZ) == (ZZ(1), f_2)
+    assert dmp_ground_primitive(
+        dmp_mul_ground(f_2, ZZ(4), 2, ZZ), 2, ZZ) == (ZZ(4), f_2)
+
+    assert dmp_ground_primitive(f_3, 2, ZZ) == (ZZ(1), f_3)
+    assert dmp_ground_primitive(
+        dmp_mul_ground(f_3, ZZ(5), 2, ZZ), 2, ZZ) == (ZZ(5), f_3)
+
+    assert dmp_ground_primitive(f_4, 2, ZZ) == (ZZ(1), f_4)
+    assert dmp_ground_primitive(
+        dmp_mul_ground(f_4, ZZ(6), 2, ZZ), 2, ZZ) == (ZZ(6), f_4)
+
+    assert dmp_ground_primitive(f_5, 2, ZZ) == (ZZ(1), f_5)
+    assert dmp_ground_primitive(
+        dmp_mul_ground(f_5, ZZ(7), 2, ZZ), 2, ZZ) == (ZZ(7), f_5)
+
+    assert dmp_ground_primitive(f_6, 3, ZZ) == (ZZ(1), f_6)
+    assert dmp_ground_primitive(
+        dmp_mul_ground(f_6, ZZ(8), 3, ZZ), 3, ZZ) == (ZZ(8), f_6)
+
+    assert dmp_ground_primitive([[ZZ(2)]], 1, ZZ) == (ZZ(2), [[ZZ(1)]])
+    assert dmp_ground_primitive([[QQ(2)]], 1, QQ) == (QQ(2), [[QQ(1)]])
+
+    assert dmp_ground_primitive(
+        [[QQ(2, 3)], [QQ(4, 9)]], 1, QQ) == (QQ(2, 9), [[QQ(3)], [QQ(2)]])
+    assert dmp_ground_primitive(
+        [[QQ(2, 3)], [QQ(4, 5)]], 1, QQ) == (QQ(2, 15), [[QQ(5)], [QQ(6)]])
+
+
+def test_dup_extract():
+    f = dup_normal([2930944, 0, 2198208, 0, 549552, 0, 45796], ZZ)
+    g = dup_normal([17585664, 0, 8792832, 0, 1099104, 0], ZZ)
+
+    F = dup_normal([64, 0, 48, 0, 12, 0, 1], ZZ)
+    G = dup_normal([384, 0, 192, 0, 24, 0], ZZ)
+
+    assert dup_extract(f, g, ZZ) == (45796, F, G)
+
+
+def test_dmp_ground_extract():
+    f = dmp_normal(
+        [[2930944], [], [2198208], [], [549552], [], [45796]], 1, ZZ)
+    g = dmp_normal([[17585664], [], [8792832], [], [1099104], []], 1, ZZ)
+
+    F = dmp_normal([[64], [], [48], [], [12], [], [1]], 1, ZZ)
+    G = dmp_normal([[384], [], [192], [], [24], []], 1, ZZ)
+
+    assert dmp_ground_extract(f, g, 1, ZZ) == (45796, F, G)
+
+
+def test_dup_real_imag():
+    assert dup_real_imag([], ZZ) == ([[]], [[]])
+    assert dup_real_imag([1], ZZ) == ([[1]], [[]])
+
+    assert dup_real_imag([1, 1], ZZ) == ([[1], [1]], [[1, 0]])
+    assert dup_real_imag([1, 2], ZZ) == ([[1], [2]], [[1, 0]])
+
+    assert dup_real_imag(
+        [1, 2, 3], ZZ) == ([[1], [2], [-1, 0, 3]], [[2, 0], [2, 0]])
+
+    assert dup_real_imag([ZZ(1), ZZ(0), ZZ(1), ZZ(3)], ZZ) == (
+        [[ZZ(1)], [], [ZZ(-3), ZZ(0), ZZ(1)], [ZZ(3)]],
+        [[ZZ(3), ZZ(0)], [], [ZZ(-1), ZZ(0), ZZ(1), ZZ(0)]]
+    )
+
+    raises(DomainError, lambda: dup_real_imag([EX(1), EX(2)], EX))
+
+
+
+def test_dup_mirror():
+    assert dup_mirror([], ZZ) == []
+    assert dup_mirror([1], ZZ) == [1]
+
+    assert dup_mirror([1, 2, 3, 4, 5], ZZ) == [1, -2, 3, -4, 5]
+    assert dup_mirror([1, 2, 3, 4, 5, 6], ZZ) == [-1, 2, -3, 4, -5, 6]
+
+
+def test_dup_scale():
+    assert dup_scale([], -1, ZZ) == []
+    assert dup_scale([1], -1, ZZ) == [1]
+
+    assert dup_scale([1, 2, 3, 4, 5], -1, ZZ) == [1, -2, 3, -4, 5]
+    assert dup_scale([1, 2, 3, 4, 5], -7, ZZ) == [2401, -686, 147, -28, 5]
+
+
+def test_dup_shift():
+    assert dup_shift([], 1, ZZ) == []
+    assert dup_shift([1], 1, ZZ) == [1]
+
+    assert dup_shift([1, 2, 3, 4, 5], 1, ZZ) == [1, 6, 15, 20, 15]
+    assert dup_shift([1, 2, 3, 4, 5], 7, ZZ) == [1, 30, 339, 1712, 3267]
+
+
+def test_dmp_shift():
+    assert dmp_shift([ZZ(1), ZZ(2)], [ZZ(1)], 0, ZZ) == [ZZ(1), ZZ(3)]
+
+    assert dmp_shift([[]], [ZZ(1), ZZ(2)], 1, ZZ) == [[]]
+
+    xy = [[ZZ(1), ZZ(0)], []]               # x*y
+    x1y2 = [[ZZ(1), ZZ(2)], [ZZ(1), ZZ(2)]] # (x+1)*(y+2)
+    assert dmp_shift(xy, [ZZ(1), ZZ(2)], 1, ZZ) == x1y2
+
+
+def test_dup_transform():
+    assert dup_transform([], [], [1, 1], ZZ) == []
+    assert dup_transform([], [1], [1, 1], ZZ) == []
+    assert dup_transform([], [1, 2], [1, 1], ZZ) == []
+
+    assert dup_transform([6, -5, 4, -3, 17], [1, -3, 4], [2, -3], ZZ) == \
+        [6, -82, 541, -2205, 6277, -12723, 17191, -13603, 4773]
+
+
+def test_dup_compose():
+    assert dup_compose([], [], ZZ) == []
+    assert dup_compose([], [1], ZZ) == []
+    assert dup_compose([], [1, 2], ZZ) == []
+
+    assert dup_compose([1], [], ZZ) == [1]
+
+    assert dup_compose([1, 2, 0], [], ZZ) == []
+    assert dup_compose([1, 2, 1], [], ZZ) == [1]
+
+    assert dup_compose([1, 2, 1], [1], ZZ) == [4]
+    assert dup_compose([1, 2, 1], [7], ZZ) == [64]
+
+    assert dup_compose([1, 2, 1], [1, -1], ZZ) == [1, 0, 0]
+    assert dup_compose([1, 2, 1], [1, 1], ZZ) == [1, 4, 4]
+    assert dup_compose([1, 2, 1], [1, 2, 1], ZZ) == [1, 4, 8, 8, 4]
+
+
+def test_dmp_compose():
+    assert dmp_compose([1, 2, 1], [1, 2, 1], 0, ZZ) == [1, 4, 8, 8, 4]
+
+    assert dmp_compose([[[]]], [[[]]], 2, ZZ) == [[[]]]
+    assert dmp_compose([[[]]], [[[1]]], 2, ZZ) == [[[]]]
+    assert dmp_compose([[[]]], [[[1]], [[2]]], 2, ZZ) == [[[]]]
+
+    assert dmp_compose([[[1]]], [], 2, ZZ) == [[[1]]]
+
+    assert dmp_compose([[1], [2], [ ]], [[]], 1, ZZ) == [[]]
+    assert dmp_compose([[1], [2], [1]], [[]], 1, ZZ) == [[1]]
+
+    assert dmp_compose([[1], [2], [1]], [[1]], 1, ZZ) == [[4]]
+    assert dmp_compose([[1], [2], [1]], [[7]], 1, ZZ) == [[64]]
+
+    assert dmp_compose([[1], [2], [1]], [[1], [-1]], 1, ZZ) == [[1], [ ], [ ]]
+    assert dmp_compose([[1], [2], [1]], [[1], [ 1]], 1, ZZ) == [[1], [4], [4]]
+
+    assert dmp_compose(
+        [[1], [2], [1]], [[1], [2], [1]], 1, ZZ) == [[1], [4], [8], [8], [4]]
+
+
+def test_dup_decompose():
+    assert dup_decompose([1], ZZ) == [[1]]
+
+    assert dup_decompose([1, 0], ZZ) == [[1, 0]]
+    assert dup_decompose([1, 0, 0, 0], ZZ) == [[1, 0, 0, 0]]
+
+    assert dup_decompose([1, 0, 0, 0, 0], ZZ) == [[1, 0, 0], [1, 0, 0]]
+    assert dup_decompose(
+        [1, 0, 0, 0, 0, 0, 0], ZZ) == [[1, 0, 0, 0], [1, 0, 0]]
+
+    assert dup_decompose([7, 0, 0, 0, 1], ZZ) == [[7, 0, 1], [1, 0, 0]]
+    assert dup_decompose([4, 0, 3, 0, 2], ZZ) == [[4, 3, 2], [1, 0, 0]]
+
+    f = [1, 0, 20, 0, 150, 0, 500, 0, 625, -2, 0, -10, 9]
+
+    assert dup_decompose(f, ZZ) == [[1, 0, 0, -2, 9], [1, 0, 5, 0]]
+
+    f = [2, 0, 40, 0, 300, 0, 1000, 0, 1250, -4, 0, -20, 18]
+
+    assert dup_decompose(f, ZZ) == [[2, 0, 0, -4, 18], [1, 0, 5, 0]]
+
+    f = [1, 0, 20, -8, 150, -120, 524, -600, 865, -1034, 600, -170, 29]
+
+    assert dup_decompose(f, ZZ) == [[1, -8, 24, -34, 29], [1, 0, 5, 0]]
+
+    R, t = ring("t", ZZ)
+    f = [6*t**2 - 42,
+         48*t**2 + 96,
+         144*t**2 + 648*t + 288,
+         624*t**2 + 864*t + 384,
+         108*t**3 + 312*t**2 + 432*t + 192]
+
+    assert dup_decompose(f, R.to_domain()) == [f]
+
+
+def test_dmp_lift():
+    q = [QQ(1, 1), QQ(0, 1), QQ(1, 1)]
+
+    f_a = [ANP([QQ(1, 1)], q, QQ), ANP([], q, QQ), ANP([], q, QQ),
+         ANP([QQ(1, 1), QQ(0, 1)], q, QQ), ANP([QQ(17, 1), QQ(0, 1)], q, QQ)]
+
+    f_lift = QQ.map([1, 0, 0, 0, 0, 0, 1, 34, 289])
+
+    assert dmp_lift(f_a, 0, QQ.algebraic_field(I)) == f_lift
+
+    f_g = [QQ_I(1), QQ_I(0), QQ_I(0), QQ_I(0, 1), QQ_I(0, 17)]
+
+    assert dmp_lift(f_g, 0, QQ_I) == f_lift
+
+    raises(DomainError, lambda: dmp_lift([EX(1), EX(2)], 0, EX))
+
+
+def test_dup_sign_variations():
+    assert dup_sign_variations([], ZZ) == 0
+    assert dup_sign_variations([1, 0], ZZ) == 0
+    assert dup_sign_variations([1, 0, 2], ZZ) == 0
+    assert dup_sign_variations([1, 0, 3, 0], ZZ) == 0
+    assert dup_sign_variations([1, 0, 4, 0, 5], ZZ) == 0
+
+    assert dup_sign_variations([-1, 0, 2], ZZ) == 1
+    assert dup_sign_variations([-1, 0, 3, 0], ZZ) == 1
+    assert dup_sign_variations([-1, 0, 4, 0, 5], ZZ) == 1
+
+    assert dup_sign_variations([-1, -4, -5], ZZ) == 0
+    assert dup_sign_variations([ 1, -4, -5], ZZ) == 1
+    assert dup_sign_variations([ 1, 4, -5], ZZ) == 1
+    assert dup_sign_variations([ 1, -4, 5], ZZ) == 2
+    assert dup_sign_variations([-1, 4, -5], ZZ) == 2
+    assert dup_sign_variations([-1, 4, 5], ZZ) == 1
+    assert dup_sign_variations([-1, -4, 5], ZZ) == 1
+    assert dup_sign_variations([ 1, 4, 5], ZZ) == 0
+
+    assert dup_sign_variations([-1, 0, -4, 0, -5], ZZ) == 0
+    assert dup_sign_variations([ 1, 0, -4, 0, -5], ZZ) == 1
+    assert dup_sign_variations([ 1, 0, 4, 0, -5], ZZ) == 1
+    assert dup_sign_variations([ 1, 0, -4, 0, 5], ZZ) == 2
+    assert dup_sign_variations([-1, 0, 4, 0, -5], ZZ) == 2
+    assert dup_sign_variations([-1, 0, 4, 0, 5], ZZ) == 1
+    assert dup_sign_variations([-1, 0, -4, 0, 5], ZZ) == 1
+    assert dup_sign_variations([ 1, 0, 4, 0, 5], ZZ) == 0
+
+
+def test_dup_clear_denoms():
+    assert dup_clear_denoms([], QQ, ZZ) == (ZZ(1), [])
+
+    assert dup_clear_denoms([QQ(1)], QQ, ZZ) == (ZZ(1), [QQ(1)])
+    assert dup_clear_denoms([QQ(7)], QQ, ZZ) == (ZZ(1), [QQ(7)])
+
+    assert dup_clear_denoms([QQ(7, 3)], QQ) == (ZZ(3), [QQ(7)])
+    assert dup_clear_denoms([QQ(7, 3)], QQ, ZZ) == (ZZ(3), [QQ(7)])
+
+    assert dup_clear_denoms(
+        [QQ(3), QQ(1), QQ(0)], QQ, ZZ) == (ZZ(1), [QQ(3), QQ(1), QQ(0)])
+    assert dup_clear_denoms(
+        [QQ(1), QQ(1, 2), QQ(0)], QQ, ZZ) == (ZZ(2), [QQ(2), QQ(1), QQ(0)])
+
+    assert dup_clear_denoms([QQ(3), QQ(
+        1), QQ(0)], QQ, ZZ, convert=True) == (ZZ(1), [ZZ(3), ZZ(1), ZZ(0)])
+    assert dup_clear_denoms([QQ(1), QQ(
+        1, 2), QQ(0)], QQ, ZZ, convert=True) == (ZZ(2), [ZZ(2), ZZ(1), ZZ(0)])
+
+    assert dup_clear_denoms(
+        [EX(S(3)/2), EX(S(9)/4)], EX) == (EX(4), [EX(6), EX(9)])
+
+    assert dup_clear_denoms([EX(7)], EX) == (EX(1), [EX(7)])
+    assert dup_clear_denoms([EX(sin(x)/x), EX(0)], EX) == (EX(x), [EX(sin(x)), EX(0)])
+
+    F = RR.frac_field(x)
+    result = dup_clear_denoms([F(8.48717/(8.0089*x + 2.83)), F(0.0)], F)
+    assert str(result) == "(x + 0.353356890459364, [1.05971731448763, 0.0])"
+
+def test_dmp_clear_denoms():
+    assert dmp_clear_denoms([[]], 1, QQ, ZZ) == (ZZ(1), [[]])
+
+    assert dmp_clear_denoms([[QQ(1)]], 1, QQ, ZZ) == (ZZ(1), [[QQ(1)]])
+    assert dmp_clear_denoms([[QQ(7)]], 1, QQ, ZZ) == (ZZ(1), [[QQ(7)]])
+
+    assert dmp_clear_denoms([[QQ(7, 3)]], 1, QQ) == (ZZ(3), [[QQ(7)]])
+    assert dmp_clear_denoms([[QQ(7, 3)]], 1, QQ, ZZ) == (ZZ(3), [[QQ(7)]])
+
+    assert dmp_clear_denoms(
+        [[QQ(3)], [QQ(1)], []], 1, QQ, ZZ) == (ZZ(1), [[QQ(3)], [QQ(1)], []])
+    assert dmp_clear_denoms([[QQ(
+        1)], [QQ(1, 2)], []], 1, QQ, ZZ) == (ZZ(2), [[QQ(2)], [QQ(1)], []])
+
+    assert dmp_clear_denoms([QQ(3), QQ(
+        1), QQ(0)], 0, QQ, ZZ, convert=True) == (ZZ(1), [ZZ(3), ZZ(1), ZZ(0)])
+    assert dmp_clear_denoms([QQ(1), QQ(1, 2), QQ(
+        0)], 0, QQ, ZZ, convert=True) == (ZZ(2), [ZZ(2), ZZ(1), ZZ(0)])
+
+    assert dmp_clear_denoms([[QQ(3)], [QQ(
+        1)], []], 1, QQ, ZZ, convert=True) == (ZZ(1), [[QQ(3)], [QQ(1)], []])
+    assert dmp_clear_denoms([[QQ(1)], [QQ(1, 2)], []], 1, QQ, ZZ,
+                            convert=True) == (ZZ(2), [[QQ(2)], [QQ(1)], []])
+
+    assert dmp_clear_denoms(
+        [[EX(S(3)/2)], [EX(S(9)/4)]], 1, EX) == (EX(4), [[EX(6)], [EX(9)]])
+    assert dmp_clear_denoms([[EX(7)]], 1, EX) == (EX(1), [[EX(7)]])
+    assert dmp_clear_denoms([[EX(sin(x)/x), EX(0)]], 1, EX) == (EX(x), [[EX(sin(x)), EX(0)]])

.venv/lib/python3.13/site-packages/sympy/polys/tests/test_dispersion.py

@@ -0,0 +1,95 @@
+from sympy.core import Symbol, S, oo
+from sympy.functions.elementary.miscellaneous import sqrt
+from sympy.polys import poly
+from sympy.polys.dispersion import dispersion, dispersionset
+
+
+def test_dispersion():
+    x = Symbol("x")
+    a = Symbol("a")
+
+    fp = poly(S.Zero, x)
+    assert sorted(dispersionset(fp)) == [0]
+
+    fp = poly(S(2), x)
+    assert sorted(dispersionset(fp)) == [0]
+
+    fp = poly(x + 1, x)
+    assert sorted(dispersionset(fp)) == [0]
+    assert dispersion(fp) == 0
+
+    fp = poly((x + 1)*(x + 2), x)
+    assert sorted(dispersionset(fp)) == [0, 1]
+    assert dispersion(fp) == 1
+
+    fp = poly(x*(x + 3), x)
+    assert sorted(dispersionset(fp)) == [0, 3]
+    assert dispersion(fp) == 3
+
+    fp = poly((x - 3)*(x + 3), x)
+    assert sorted(dispersionset(fp)) == [0, 6]
+    assert dispersion(fp) == 6
+
+    fp = poly(x**4 - 3*x**2 + 1, x)
+    gp = fp.shift(-3)
+    assert sorted(dispersionset(fp, gp)) == [2, 3, 4]
+    assert dispersion(fp, gp) == 4
+    assert sorted(dispersionset(gp, fp)) == []
+    assert dispersion(gp, fp) is -oo
+
+    fp = poly(x*(3*x**2+a)*(x-2536)*(x**3+a), x)
+    gp = fp.as_expr().subs(x, x-345).as_poly(x)
+    assert sorted(dispersionset(fp, gp)) == [345, 2881]
+    assert sorted(dispersionset(gp, fp)) == [2191]
+
+    gp = poly((x-2)**2*(x-3)**3*(x-5)**3, x)
+    assert sorted(dispersionset(gp)) == [0, 1, 2, 3]
+    assert sorted(dispersionset(gp, (gp+4)**2)) == [1, 2]
+
+    fp = poly(x*(x+2)*(x-1), x)
+    assert sorted(dispersionset(fp)) == [0, 1, 2, 3]
+
+    fp = poly(x**2 + sqrt(5)*x - 1, x, domain='QQ<sqrt(5)>')
+    gp = poly(x**2 + (2 + sqrt(5))*x + sqrt(5), x, domain='QQ<sqrt(5)>')
+    assert sorted(dispersionset(fp, gp)) == [2]
+    assert sorted(dispersionset(gp, fp)) == [1, 4]
+
+    # There are some difficulties if we compute over Z[a]
+    # and alpha happens to lie in Z[a] instead of simply Z.
+    # Hence we can not decide if alpha is indeed integral
+    # in general.
+
+    fp = poly(4*x**4 + (4*a + 8)*x**3 + (a**2 + 6*a + 4)*x**2 + (a**2 + 2*a)*x, x)
+    assert sorted(dispersionset(fp)) == [0, 1]
+
+    # For any specific value of a, the dispersion is 3*a
+    # but the algorithm can not find this in general.
+    # This is the point where the resultant based Ansatz
+    # is superior to the current one.
+    fp = poly(a**2*x**3 + (a**3 + a**2 + a + 1)*x, x)
+    gp = fp.as_expr().subs(x, x - 3*a).as_poly(x)
+    assert sorted(dispersionset(fp, gp)) == []
+
+    fpa = fp.as_expr().subs(a, 2).as_poly(x)
+    gpa = gp.as_expr().subs(a, 2).as_poly(x)
+    assert sorted(dispersionset(fpa, gpa)) == [6]
+
+    # Work with Expr instead of Poly
+    f = (x + 1)*(x + 2)
+    assert sorted(dispersionset(f)) == [0, 1]
+    assert dispersion(f) == 1
+
+    f = x**4 - 3*x**2 + 1
+    g = x**4 - 12*x**3 + 51*x**2 - 90*x + 55
+    assert sorted(dispersionset(f, g)) == [2, 3, 4]
+    assert dispersion(f, g) == 4
+
+    # Work with Expr and specify a generator
+    f = (x + 1)*(x + 2)
+    assert sorted(dispersionset(f, None, x)) == [0, 1]
+    assert dispersion(f, None, x) == 1
+
+    f = x**4 - 3*x**2 + 1
+    g = x**4 - 12*x**3 + 51*x**2 - 90*x + 55
+    assert sorted(dispersionset(f, g, x)) == [2, 3, 4]
+    assert dispersion(f, g, x) == 4

.venv/lib/python3.13/site-packages/sympy/polys/tests/test_distributedmodules.py

@@ -0,0 +1,208 @@
+"""Tests for sparse distributed modules. """
+
+from sympy.polys.distributedmodules import (
+    sdm_monomial_mul, sdm_monomial_deg, sdm_monomial_divides,
+    sdm_add, sdm_LM, sdm_LT, sdm_mul_term, sdm_zero, sdm_deg,
+    sdm_LC, sdm_from_dict,
+    sdm_spoly, sdm_ecart, sdm_nf_mora, sdm_groebner,
+    sdm_from_vector, sdm_to_vector, sdm_monomial_lcm
+)
+
+from sympy.polys.orderings import lex, grlex, InverseOrder
+from sympy.polys.domains import QQ
+
+from sympy.abc import x, y, z
+
+
+def test_sdm_monomial_mul():
+    assert sdm_monomial_mul((1, 1, 0), (1, 3)) == (1, 2, 3)
+
+
+def test_sdm_monomial_deg():
+    assert sdm_monomial_deg((5, 2, 1)) == 3
+
+
+def test_sdm_monomial_lcm():
+    assert sdm_monomial_lcm((1, 2, 3), (1, 5, 0)) == (1, 5, 3)
+
+
+def test_sdm_monomial_divides():
+    assert sdm_monomial_divides((1, 0, 0), (1, 0, 0)) is True
+    assert sdm_monomial_divides((1, 0, 0), (1, 2, 1)) is True
+    assert sdm_monomial_divides((5, 1, 1), (5, 2, 1)) is True
+
+    assert sdm_monomial_divides((1, 0, 0), (2, 0, 0)) is False
+    assert sdm_monomial_divides((1, 1, 0), (1, 0, 0)) is False
+    assert sdm_monomial_divides((5, 1, 2), (5, 0, 1)) is False
+
+
+def test_sdm_LC():
+    assert sdm_LC([((1, 2, 3), QQ(5))], QQ) == QQ(5)
+
+
+def test_sdm_from_dict():
+    dic = {(1, 2, 1, 1): QQ(1), (1, 1, 2, 1): QQ(1), (1, 0, 2, 1): QQ(1),
+           (1, 0, 0, 3): QQ(1), (1, 1, 1, 0): QQ(1)}
+    assert sdm_from_dict(dic, grlex) == \
+        [((1, 2, 1, 1), QQ(1)), ((1, 1, 2, 1), QQ(1)),
+         ((1, 0, 2, 1), QQ(1)), ((1, 0, 0, 3), QQ(1)), ((1, 1, 1, 0), QQ(1))]
+
+# TODO test to_dict?
+
+
+def test_sdm_add():
+    assert sdm_add([((1, 1, 1), QQ(1))], [((2, 0, 0), QQ(1))], lex, QQ) == \
+        [((2, 0, 0), QQ(1)), ((1, 1, 1), QQ(1))]
+    assert sdm_add([((1, 1, 1), QQ(1))], [((1, 1, 1), QQ(-1))], lex, QQ) == []
+    assert sdm_add([((1, 0, 0), QQ(1))], [((1, 0, 0), QQ(2))], lex, QQ) == \
+        [((1, 0, 0), QQ(3))]
+    assert sdm_add([((1, 0, 1), QQ(1))], [((1, 1, 0), QQ(1))], lex, QQ) == \
+        [((1, 1, 0), QQ(1)), ((1, 0, 1), QQ(1))]
+
+
+def test_sdm_LM():
+    dic = {(1, 2, 3): QQ(1), (4, 0, 0): QQ(1), (4, 0, 1): QQ(1)}
+    assert sdm_LM(sdm_from_dict(dic, lex)) == (4, 0, 1)
+
+
+def test_sdm_LT():
+    dic = {(1, 2, 3): QQ(1), (4, 0, 0): QQ(2), (4, 0, 1): QQ(3)}
+    assert sdm_LT(sdm_from_dict(dic, lex)) == ((4, 0, 1), QQ(3))
+
+
+def test_sdm_mul_term():
+    assert sdm_mul_term([((1, 0, 0), QQ(1))], ((0, 0), QQ(0)), lex, QQ) == []
+    assert sdm_mul_term([], ((1, 0), QQ(1)), lex, QQ) == []
+    assert sdm_mul_term([((1, 0, 0), QQ(1))], ((1, 0), QQ(1)), lex, QQ) == \
+        [((1, 1, 0), QQ(1))]
+    f = [((2, 0, 1), QQ(4)), ((1, 1, 0), QQ(3))]
+    assert sdm_mul_term(f, ((1, 1), QQ(2)), lex, QQ) == \
+        [((2, 1, 2), QQ(8)), ((1, 2, 1), QQ(6))]
+
+
+def test_sdm_zero():
+    assert sdm_zero() == []
+
+
+def test_sdm_deg():
+    assert sdm_deg([((1, 2, 3), 1), ((10, 0, 1), 1), ((2, 3, 4), 4)]) == 7
+
+
+def test_sdm_spoly():
+    f = [((2, 1, 1), QQ(1)), ((1, 0, 1), QQ(1))]
+    g = [((2, 3, 0), QQ(1))]
+    h = [((1, 2, 3), QQ(1))]
+    assert sdm_spoly(f, h, lex, QQ) == []
+    assert sdm_spoly(f, g, lex, QQ) == [((1, 2, 1), QQ(1))]
+
+
+def test_sdm_ecart():
+    assert sdm_ecart([((1, 2, 3), 1), ((1, 0, 1), 1)]) == 0
+    assert sdm_ecart([((2, 2, 1), 1), ((1, 5, 1), 1)]) == 3
+
+
+def test_sdm_nf_mora():
+    f = sdm_from_dict({(1, 2, 1, 1): QQ(1), (1, 1, 2, 1): QQ(1),
+                (1, 0, 2, 1): QQ(1), (1, 0, 0, 3): QQ(1), (1, 1, 1, 0): QQ(1)},
+        grlex)
+    f1 = sdm_from_dict({(1, 1, 1, 0): QQ(1), (1, 0, 2, 0): QQ(1),
+                        (1, 0, 0, 0): QQ(-1)}, grlex)
+    f2 = sdm_from_dict({(1, 1, 1, 0): QQ(1)}, grlex)
+    (id0, id1, id2) = [sdm_from_dict({(i, 0, 0, 0): QQ(1)}, grlex)
+                       for i in range(3)]
+
+    assert sdm_nf_mora(f, [f1, f2], grlex, QQ, phantom=(id0, [id1, id2])) == \
+        ([((1, 0, 2, 1), QQ(1)), ((1, 0, 0, 3), QQ(1)), ((1, 1, 1, 0), QQ(1)),
+          ((1, 1, 0, 1), QQ(1))],
+         [((1, 1, 0, 1), QQ(-1)), ((0, 0, 0, 0), QQ(1))])
+    assert sdm_nf_mora(f, [f2, f1], grlex, QQ, phantom=(id0, [id2, id1])) == \
+        ([((1, 0, 2, 1), QQ(1)), ((1, 0, 0, 3), QQ(1)), ((1, 1, 1, 0), QQ(1))],
+         [((2, 1, 0, 1), QQ(-1)), ((2, 0, 1, 1), QQ(-1)), ((0, 0, 0, 0), QQ(1))])
+
+    f = sdm_from_vector([x*z, y**2 + y*z - z, y], lex, QQ, gens=[x, y, z])
+    f1 = sdm_from_vector([x, y, 1], lex, QQ, gens=[x, y, z])
+    f2 = sdm_from_vector([x*y, z, z**2], lex, QQ, gens=[x, y, z])
+    assert sdm_nf_mora(f, [f1, f2], lex, QQ) == \
+        sdm_nf_mora(f, [f2, f1], lex, QQ) == \
+        [((1, 0, 1, 1), QQ(1)), ((1, 0, 0, 1), QQ(-1)), ((0, 1, 1, 0), QQ(-1)),
+         ((0, 1, 0, 1), QQ(1))]
+
+
+def test_conversion():
+    f = [x**2 + y**2, 2*z]
+    g = [((1, 0, 0, 1), QQ(2)), ((0, 2, 0, 0), QQ(1)), ((0, 0, 2, 0), QQ(1))]
+    assert sdm_to_vector(g, [x, y, z], QQ) == f
+    assert sdm_from_vector(f, lex, QQ) == g
+    assert sdm_from_vector(
+        [x, 1], lex, QQ) == [((1, 0), QQ(1)), ((0, 1), QQ(1))]
+    assert sdm_to_vector([((1, 1, 0, 0), 1)], [x, y, z], QQ, n=3) == [0, x, 0]
+    assert sdm_from_vector([0, 0], lex, QQ, gens=[x, y]) == sdm_zero()
+
+
+def test_nontrivial():
+    gens = [x, y, z]
+
+    def contains(I, f):
+        S = [sdm_from_vector([g], lex, QQ, gens=gens) for g in I]
+        G = sdm_groebner(S, sdm_nf_mora, lex, QQ)
+        return sdm_nf_mora(sdm_from_vector([f], lex, QQ, gens=gens),
+                           G, lex, QQ) == sdm_zero()
+
+    assert contains([x, y], x)
+    assert contains([x, y], x + y)
+    assert not contains([x, y], 1)
+    assert not contains([x, y], z)
+    assert contains([x**2 + y, x**2 + x], x - y)
+    assert not contains([x + y + z, x*y + x*z + y*z, x*y*z], x**2)
+    assert contains([x + y + z, x*y + x*z + y*z, x*y*z], x**3)
+    assert contains([x + y + z, x*y + x*z + y*z, x*y*z], x**4)
+    assert not contains([x + y + z, x*y + x*z + y*z, x*y*z], x*y**2)
+    assert contains([x + y + z, x*y + x*z + y*z, x*y*z], x**4 + y**3 + 2*z*y*x)
+    assert contains([x + y + z, x*y + x*z + y*z, x*y*z], x*y*z)
+    assert contains([x, 1 + x + y, 5 - 7*y], 1)
+    assert contains(
+        [x**3 + y**3, y**3 + z**3, z**3 + x**3, x**2*y + x**2*z + y**2*z],
+        x**3)
+    assert not contains(
+        [x**3 + y**3, y**3 + z**3, z**3 + x**3, x**2*y + x**2*z + y**2*z],
+        x**2 + y**2)
+
+    # compare local order
+    assert not contains([x*(1 + x + y), y*(1 + z)], x)
+    assert not contains([x*(1 + x + y), y*(1 + z)], x + y)
+
+
+def test_local():
+    igrlex = InverseOrder(grlex)
+    gens = [x, y, z]
+
+    def contains(I, f):
+        S = [sdm_from_vector([g], igrlex, QQ, gens=gens) for g in I]
+        G = sdm_groebner(S, sdm_nf_mora, igrlex, QQ)
+        return sdm_nf_mora(sdm_from_vector([f], lex, QQ, gens=gens),
+                           G, lex, QQ) == sdm_zero()
+    assert contains([x, y], x)
+    assert contains([x, y], x + y)
+    assert not contains([x, y], 1)
+    assert not contains([x, y], z)
+    assert contains([x**2 + y, x**2 + x], x - y)
+    assert not contains([x + y + z, x*y + x*z + y*z, x*y*z], x**2)
+    assert contains([x*(1 + x + y), y*(1 + z)], x)
+    assert contains([x*(1 + x + y), y*(1 + z)], x + y)
+
+
+def test_uncovered_line():
+    gens = [x, y]
+    f1 = sdm_zero()
+    f2 = sdm_from_vector([x, 0], lex, QQ, gens=gens)
+    f3 = sdm_from_vector([0, y], lex, QQ, gens=gens)
+
+    assert sdm_spoly(f1, f2, lex, QQ) == sdm_zero()
+    assert sdm_spoly(f3, f2, lex, QQ) == sdm_zero()
+
+
+def test_chain_criterion():
+    gens = [x]
+    f1 = sdm_from_vector([1, x], grlex, QQ, gens=gens)
+    f2 = sdm_from_vector([0, x - 2], grlex, QQ, gens=gens)
+    assert len(sdm_groebner([f1, f2], sdm_nf_mora, grlex, QQ)) == 2

.venv/lib/python3.13/site-packages/sympy/polys/tests/test_euclidtools.py

@@ -0,0 +1,712 @@
+"""Tests for Euclidean algorithms, GCDs, LCMs and polynomial remainder sequences. """
+
+from sympy.polys.rings import ring
+from sympy.polys.domains import ZZ, QQ, RR
+
+from sympy.polys.specialpolys import (
+    f_polys,
+    dmp_fateman_poly_F_1,
+    dmp_fateman_poly_F_2,
+    dmp_fateman_poly_F_3)
+
+f_0, f_1, f_2, f_3, f_4, f_5, f_6 = f_polys()
+
+def test_dup_gcdex():
+    R, x = ring("x", QQ)
+
+    f = x**4 - 2*x**3 - 6*x**2 + 12*x + 15
+    g = x**3 + x**2 - 4*x - 4
+
+    s = -QQ(1,5)*x + QQ(3,5)
+    t = QQ(1,5)*x**2 - QQ(6,5)*x + 2
+    h = x + 1
+
+    assert R.dup_half_gcdex(f, g) == (s, h)
+    assert R.dup_gcdex(f, g) == (s, t, h)
+
+    f = x**4 + 4*x**3 - x + 1
+    g = x**3 - x + 1
+
+    s, t, h = R.dup_gcdex(f, g)
+    S, T, H = R.dup_gcdex(g, f)
+
+    assert R.dup_add(R.dup_mul(s, f),
+                     R.dup_mul(t, g)) == h
+    assert R.dup_add(R.dup_mul(S, g),
+                     R.dup_mul(T, f)) == H
+
+    f = 2*x
+    g = x**2 - 16
+
+    s = QQ(1,32)*x
+    t = -QQ(1,16)
+    h = 1
+
+    assert R.dup_half_gcdex(f, g) == (s, h)
+    assert R.dup_gcdex(f, g) == (s, t, h)
+
+
+def test_dup_invert():
+    R, x = ring("x", QQ)
+    assert R.dup_invert(2*x, x**2 - 16) == QQ(1,32)*x
+
+
+def test_dup_euclidean_prs():
+    R, x = ring("x", QQ)
+
+    f = x**8 + x**6 - 3*x**4 - 3*x**3 + 8*x**2 + 2*x - 5
+    g = 3*x**6 + 5*x**4 - 4*x**2 - 9*x + 21
+
+    assert R.dup_euclidean_prs(f, g) == [
+        f,
+        g,
+        -QQ(5,9)*x**4 + QQ(1,9)*x**2 - QQ(1,3),
+        -QQ(117,25)*x**2 - 9*x + QQ(441,25),
+        QQ(233150,19773)*x - QQ(102500,6591),
+        -QQ(1288744821,543589225)]
+
+
+def test_dup_primitive_prs():
+    R, x = ring("x", ZZ)
+
+    f = x**8 + x**6 - 3*x**4 - 3*x**3 + 8*x**2 + 2*x - 5
+    g = 3*x**6 + 5*x**4 - 4*x**2 - 9*x + 21
+
+    assert R.dup_primitive_prs(f, g) == [
+        f,
+        g,
+        -5*x**4 + x**2 - 3,
+        13*x**2 + 25*x - 49,
+        4663*x - 6150,
+        1]
+
+
+def test_dup_subresultants():
+    R, x = ring("x", ZZ)
+
+    assert R.dup_resultant(0, 0) == 0
+
+    assert R.dup_resultant(1, 0) == 0
+    assert R.dup_resultant(0, 1) == 0
+
+    f = x**8 + x**6 - 3*x**4 - 3*x**3 + 8*x**2 + 2*x - 5
+    g = 3*x**6 + 5*x**4 - 4*x**2 - 9*x + 21
+
+    a = 15*x**4 - 3*x**2 + 9
+    b = 65*x**2 + 125*x - 245
+    c = 9326*x - 12300
+    d = 260708
+
+    assert R.dup_subresultants(f, g) == [f, g, a, b, c, d]
+    assert R.dup_resultant(f, g) == R.dup_LC(d)
+
+    f = x**2 - 2*x + 1
+    g = x**2 - 1
+
+    a = 2*x - 2
+
+    assert R.dup_subresultants(f, g) == [f, g, a]
+    assert R.dup_resultant(f, g) == 0
+
+    f = x**2 + 1
+    g = x**2 - 1
+
+    a = -2
+
+    assert R.dup_subresultants(f, g) == [f, g, a]
+    assert R.dup_resultant(f, g) == 4
+
+    f = x**2 - 1
+    g = x**3 - x**2 + 2
+
+    assert R.dup_resultant(f, g) == 0
+
+    f = 3*x**3 - x
+    g = 5*x**2 + 1
+
+    assert R.dup_resultant(f, g) == 64
+
+    f = x**2 - 2*x + 7
+    g = x**3 - x + 5
+
+    assert R.dup_resultant(f, g) == 265
+
+    f = x**3 - 6*x**2 + 11*x - 6
+    g = x**3 - 15*x**2 + 74*x - 120
+
+    assert R.dup_resultant(f, g) == -8640
+
+    f = x**3 - 6*x**2 + 11*x - 6
+    g = x**3 - 10*x**2 + 29*x - 20
+
+    assert R.dup_resultant(f, g) == 0
+
+    f = x**3 - 1
+    g = x**3 + 2*x**2 + 2*x - 1
+
+    assert R.dup_resultant(f, g) == 16
+
+    f = x**8 - 2
+    g = x - 1
+
+    assert R.dup_resultant(f, g) == -1
+
+
+def test_dmp_subresultants():
+    R, x, y = ring("x,y", ZZ)
+
+    assert R.dmp_resultant(0, 0) == 0
+    assert R.dmp_prs_resultant(0, 0)[0] == 0
+    assert R.dmp_zz_collins_resultant(0, 0) == 0
+    assert R.dmp_qq_collins_resultant(0, 0) == 0
+
+    assert R.dmp_resultant(1, 0) == 0
+    assert R.dmp_resultant(1, 0) == 0
+    assert R.dmp_resultant(1, 0) == 0
+
+    assert R.dmp_resultant(0, 1) == 0
+    assert R.dmp_prs_resultant(0, 1)[0] == 0
+    assert R.dmp_zz_collins_resultant(0, 1) == 0
+    assert R.dmp_qq_collins_resultant(0, 1) == 0
+
+    f = 3*x**2*y - y**3 - 4
+    g = x**2 + x*y**3 - 9
+
+    a = 3*x*y**4 + y**3 - 27*y + 4
+    b = -3*y**10 - 12*y**7 + y**6 - 54*y**4 + 8*y**3 + 729*y**2 - 216*y + 16
+
+    r = R.dmp_LC(b)
+
+    assert R.dmp_subresultants(f, g) == [f, g, a, b]
+
+    assert R.dmp_resultant(f, g) == r
+    assert R.dmp_prs_resultant(f, g)[0] == r
+    assert R.dmp_zz_collins_resultant(f, g) == r
+    assert R.dmp_qq_collins_resultant(f, g) == r
+
+    f = -x**3 + 5
+    g = 3*x**2*y + x**2
+
+    a = 45*y**2 + 30*y + 5
+    b = 675*y**3 + 675*y**2 + 225*y + 25
+
+    r = R.dmp_LC(b)
+
+    assert R.dmp_subresultants(f, g) == [f, g, a]
+    assert R.dmp_resultant(f, g) == r
+    assert R.dmp_prs_resultant(f, g)[0] == r
+    assert R.dmp_zz_collins_resultant(f, g) == r
+    assert R.dmp_qq_collins_resultant(f, g) == r
+
+    R, x, y, z, u, v = ring("x,y,z,u,v", ZZ)
+
+    f = 6*x**2 - 3*x*y - 2*x*z + y*z
+    g = x**2 - x*u - x*v + u*v
+
+    r = y**2*z**2 - 3*y**2*z*u - 3*y**2*z*v + 9*y**2*u*v - 2*y*z**2*u \
+      - 2*y*z**2*v + 6*y*z*u**2 + 12*y*z*u*v + 6*y*z*v**2 - 18*y*u**2*v \
+      - 18*y*u*v**2 + 4*z**2*u*v - 12*z*u**2*v - 12*z*u*v**2 + 36*u**2*v**2
+
+    assert R.dmp_zz_collins_resultant(f, g) == r.drop(x)
+
+    R, x, y, z, u, v = ring("x,y,z,u,v", QQ)
+
+    f = x**2 - QQ(1,2)*x*y - QQ(1,3)*x*z + QQ(1,6)*y*z
+    g = x**2 - x*u - x*v + u*v
+
+    r = QQ(1,36)*y**2*z**2 - QQ(1,12)*y**2*z*u - QQ(1,12)*y**2*z*v + QQ(1,4)*y**2*u*v \
+      - QQ(1,18)*y*z**2*u - QQ(1,18)*y*z**2*v + QQ(1,6)*y*z*u**2 + QQ(1,3)*y*z*u*v \
+      + QQ(1,6)*y*z*v**2 - QQ(1,2)*y*u**2*v - QQ(1,2)*y*u*v**2 + QQ(1,9)*z**2*u*v \
+      - QQ(1,3)*z*u**2*v - QQ(1,3)*z*u*v**2 + u**2*v**2
+
+    assert R.dmp_qq_collins_resultant(f, g) == r.drop(x)
+
+    Rt, t = ring("t", ZZ)
+    Rx, x = ring("x", Rt)
+
+    f = x**6 - 5*x**4 + 5*x**2 + 4
+    g = -6*t*x**5 + x**4 + 20*t*x**3 - 3*x**2 - 10*t*x + 6
+
+    assert Rx.dup_resultant(f, g) == 2930944*t**6 + 2198208*t**4 + 549552*t**2 + 45796
+
+
+def test_dup_discriminant():
+    R, x = ring("x", ZZ)
+
+    assert R.dup_discriminant(0) == 0
+    assert R.dup_discriminant(x) == 1
+
+    assert R.dup_discriminant(x**3 + 3*x**2 + 9*x - 13) == -11664
+    assert R.dup_discriminant(5*x**5 + x**3 + 2) == 31252160
+    assert R.dup_discriminant(x**4 + 2*x**3 + 6*x**2 - 22*x + 13) == 0
+    assert R.dup_discriminant(12*x**7 + 15*x**4 + 30*x**3 + x**2 + 1) == -220289699947514112
+
+
+def test_dmp_discriminant():
+    R, x = ring("x", ZZ)
+
+    assert R.dmp_discriminant(0) == 0
+
+    R, x, y = ring("x,y", ZZ)
+
+    assert R.dmp_discriminant(0) == 0
+    assert R.dmp_discriminant(y) == 0
+
+    assert R.dmp_discriminant(x**3 + 3*x**2 + 9*x - 13) == -11664
+    assert R.dmp_discriminant(5*x**5 + x**3 + 2) == 31252160
+    assert R.dmp_discriminant(x**4 + 2*x**3 + 6*x**2 - 22*x + 13) == 0
+    assert R.dmp_discriminant(12*x**7 + 15*x**4 + 30*x**3 + x**2 + 1) == -220289699947514112
+
+    assert R.dmp_discriminant(x**2*y + 2*y) == (-8*y**2).drop(x)
+    assert R.dmp_discriminant(x*y**2 + 2*x) == 1
+
+    R, x, y, z = ring("x,y,z", ZZ)
+    assert R.dmp_discriminant(x*y + z) == 1
+
+    R, x, y, z, u = ring("x,y,z,u", ZZ)
+    assert R.dmp_discriminant(x**2*y + x*z + u) == (-4*y*u + z**2).drop(x)
+
+    R, x, y, z, u, v = ring("x,y,z,u,v", ZZ)
+    assert R.dmp_discriminant(x**3*y + x**2*z + x*u + v) == \
+        (-27*y**2*v**2 + 18*y*z*u*v - 4*y*u**3 - 4*z**3*v + z**2*u**2).drop(x)
+
+
+def test_dup_gcd():
+    R, x = ring("x", ZZ)
+
+    f, g = 0, 0
+    assert R.dup_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (0, 0, 0)
+
+    f, g = 2, 0
+    assert R.dup_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (2, 1, 0)
+
+    f, g = -2, 0
+    assert R.dup_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (2, -1, 0)
+
+    f, g = 0, -2
+    assert R.dup_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (2, 0, -1)
+
+    f, g = 0, 2*x + 4
+    assert R.dup_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (2*x + 4, 0, 1)
+
+    f, g = 2*x + 4, 0
+    assert R.dup_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (2*x + 4, 1, 0)
+
+    f, g = 2, 2
+    assert R.dup_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (2, 1, 1)
+
+    f, g = -2, 2
+    assert R.dup_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (2, -1, 1)
+
+    f, g = 2, -2
+    assert R.dup_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (2, 1, -1)
+
+    f, g = -2, -2
+    assert R.dup_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (2, -1, -1)
+
+    f, g = x**2 + 2*x + 1, 1
+    assert R.dup_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (1, x**2 + 2*x + 1, 1)
+
+    f, g = x**2 + 2*x + 1, 2
+    assert R.dup_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (1, x**2 + 2*x + 1, 2)
+
+    f, g = 2*x**2 + 4*x + 2, 2
+    assert R.dup_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (2, x**2 + 2*x + 1, 1)
+
+    f, g = 2, 2*x**2 + 4*x + 2
+    assert R.dup_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (2, 1, x**2 + 2*x + 1)
+
+    f, g = 2*x**2 + 4*x + 2, x + 1
+    assert R.dup_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (x + 1, 2*x + 2, 1)
+
+    f, g = x + 1, 2*x**2 + 4*x + 2
+    assert R.dup_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (x + 1, 1, 2*x + 2)
+
+    f, g = x - 31, x
+    assert R.dup_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (1, f, g)
+
+    f = x**4 + 8*x**3 + 21*x**2 + 22*x + 8
+    g = x**3 + 6*x**2 + 11*x + 6
+
+    h = x**2 + 3*x + 2
+
+    cff = x**2 + 5*x + 4
+    cfg = x + 3
+
+    assert R.dup_zz_heu_gcd(f, g) == (h, cff, cfg)
+    assert R.dup_rr_prs_gcd(f, g) == (h, cff, cfg)
+
+    f = x**4 - 4
+    g = x**4 + 4*x**2 + 4
+
+    h = x**2 + 2
+
+    cff = x**2 - 2
+    cfg = x**2 + 2
+
+    assert R.dup_zz_heu_gcd(f, g) == (h, cff, cfg)
+    assert R.dup_rr_prs_gcd(f, g) == (h, cff, cfg)
+
+    f = x**8 + x**6 - 3*x**4 - 3*x**3 + 8*x**2 + 2*x - 5
+    g = 3*x**6 + 5*x**4 - 4*x**2 - 9*x + 21
+
+    h = 1
+
+    cff = f
+    cfg = g
+
+    assert R.dup_zz_heu_gcd(f, g) == (h, cff, cfg)
+    assert R.dup_rr_prs_gcd(f, g) == (h, cff, cfg)
+
+    R, x = ring("x", QQ)
+
+    f = x**8 + x**6 - 3*x**4 - 3*x**3 + 8*x**2 + 2*x - 5
+    g = 3*x**6 + 5*x**4 - 4*x**2 - 9*x + 21
+
+    h = 1
+
+    cff = f
+    cfg = g
+
+    assert R.dup_qq_heu_gcd(f, g) == (h, cff, cfg)
+    assert R.dup_ff_prs_gcd(f, g) == (h, cff, cfg)
+
+    R, x = ring("x", ZZ)
+
+    f = - 352518131239247345597970242177235495263669787845475025293906825864749649589178600387510272*x**49 \
+        + 46818041807522713962450042363465092040687472354933295397472942006618953623327997952*x**42 \
+        + 378182690892293941192071663536490788434899030680411695933646320291525827756032*x**35 \
+        + 112806468807371824947796775491032386836656074179286744191026149539708928*x**28 \
+        - 12278371209708240950316872681744825481125965781519138077173235712*x**21 \
+        + 289127344604779611146960547954288113529690984687482920704*x**14 \
+        + 19007977035740498977629742919480623972236450681*x**7 \
+        + 311973482284542371301330321821976049
+
+    g =   365431878023781158602430064717380211405897160759702125019136*x**21 \
+        + 197599133478719444145775798221171663643171734081650688*x**14 \
+        - 9504116979659010018253915765478924103928886144*x**7 \
+        - 311973482284542371301330321821976049
+
+    assert R.dup_zz_heu_gcd(f, R.dup_diff(f, 1))[0] == g
+    assert R.dup_rr_prs_gcd(f, R.dup_diff(f, 1))[0] == g
+
+    R, x = ring("x", QQ)
+
+    f = QQ(1,2)*x**2 + x + QQ(1,2)
+    g = QQ(1,2)*x + QQ(1,2)
+
+    h = x + 1
+
+    assert R.dup_qq_heu_gcd(f, g) == (h, g, QQ(1,2))
+    assert R.dup_ff_prs_gcd(f, g) == (h, g, QQ(1,2))
+
+    R, x = ring("x", ZZ)
+
+    f = 1317378933230047068160*x + 2945748836994210856960
+    g = 120352542776360960*x + 269116466014453760
+
+    h = 120352542776360960*x + 269116466014453760
+    cff = 10946
+    cfg = 1
+
+    assert R.dup_zz_heu_gcd(f, g) == (h, cff, cfg)
+
+
+def test_dmp_gcd():
+    R, x, y = ring("x,y", ZZ)
+
+    f, g = 0, 0
+    assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (0, 0, 0)
+
+    f, g = 2, 0
+    assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (2, 1, 0)
+
+    f, g = -2, 0
+    assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (2, -1, 0)
+
+    f, g = 0, -2
+    assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (2, 0, -1)
+
+    f, g = 0, 2*x + 4
+    assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (2*x + 4, 0, 1)
+
+    f, g = 2*x + 4, 0
+    assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (2*x + 4, 1, 0)
+
+    f, g = 2, 2
+    assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (2, 1, 1)
+
+    f, g = -2, 2
+    assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (2, -1, 1)
+
+    f, g = 2, -2
+    assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (2, 1, -1)
+
+    f, g = -2, -2
+    assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (2, -1, -1)
+
+    f, g = x**2 + 2*x + 1, 1
+    assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (1, x**2 + 2*x + 1, 1)
+
+    f, g = x**2 + 2*x + 1, 2
+    assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (1, x**2 + 2*x + 1, 2)
+
+    f, g = 2*x**2 + 4*x + 2, 2
+    assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (2, x**2 + 2*x + 1, 1)
+
+    f, g = 2, 2*x**2 + 4*x + 2
+    assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (2, 1, x**2 + 2*x + 1)
+
+    f, g = 2*x**2 + 4*x + 2, x + 1
+    assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (x + 1, 2*x + 2, 1)
+
+    f, g = x + 1, 2*x**2 + 4*x + 2
+    assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (x + 1, 1, 2*x + 2)
+
+    R, x, y, z, u = ring("x,y,z,u", ZZ)
+
+    f, g = u**2 + 2*u + 1, 2*u + 2
+    assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (u + 1, u + 1, 2)
+
+    f, g = z**2*u**2 + 2*z**2*u + z**2 + z*u + z, u**2 + 2*u + 1
+    h, cff, cfg = u + 1, z**2*u + z**2 + z, u + 1
+
+    assert R.dmp_zz_heu_gcd(f, g) == (h, cff, cfg)
+    assert R.dmp_rr_prs_gcd(f, g) == (h, cff, cfg)
+
+    assert R.dmp_zz_heu_gcd(g, f) == (h, cfg, cff)
+    assert R.dmp_rr_prs_gcd(g, f) == (h, cfg, cff)
+
+    R, x, y, z = ring("x,y,z", ZZ)
+
+    f, g, h = map(R.from_dense, dmp_fateman_poly_F_1(2, ZZ))
+    H, cff, cfg = R.dmp_zz_heu_gcd(f, g)
+
+    assert H == h and R.dmp_mul(H, cff) == f \
+                  and R.dmp_mul(H, cfg) == g
+
+    H, cff, cfg = R.dmp_rr_prs_gcd(f, g)
+
+    assert H == h and R.dmp_mul(H, cff) == f \
+                  and R.dmp_mul(H, cfg) == g
+
+    R, x, y, z, u, v = ring("x,y,z,u,v", ZZ)
+
+    f, g, h = map(R.from_dense, dmp_fateman_poly_F_1(4, ZZ))
+    H, cff, cfg = R.dmp_zz_heu_gcd(f, g)
+
+    assert H == h and R.dmp_mul(H, cff) == f \
+                  and R.dmp_mul(H, cfg) == g
+
+    R, x, y, z, u, v, a, b = ring("x,y,z,u,v,a,b", ZZ)
+
+    f, g, h = map(R.from_dense, dmp_fateman_poly_F_1(6, ZZ))
+    H, cff, cfg = R.dmp_zz_heu_gcd(f, g)
+
+    assert H == h and R.dmp_mul(H, cff) == f \
+                  and R.dmp_mul(H, cfg) == g
+
+    R, x, y, z, u, v, a, b, c, d = ring("x,y,z,u,v,a,b,c,d", ZZ)
+
+    f, g, h = map(R.from_dense, dmp_fateman_poly_F_1(8, ZZ))
+    H, cff, cfg = R.dmp_zz_heu_gcd(f, g)
+
+    assert H == h and R.dmp_mul(H, cff) == f \
+                  and R.dmp_mul(H, cfg) == g
+
+    R, x, y, z = ring("x,y,z", ZZ)
+
+    f, g, h = map(R.from_dense, dmp_fateman_poly_F_2(2, ZZ))
+    H, cff, cfg = R.dmp_zz_heu_gcd(f, g)
+
+    assert H == h and R.dmp_mul(H, cff) == f \
+                  and R.dmp_mul(H, cfg) == g
+
+    H, cff, cfg = R.dmp_rr_prs_gcd(f, g)
+
+    assert H == h and R.dmp_mul(H, cff) == f \
+                  and R.dmp_mul(H, cfg) == g
+
+    f, g, h = map(R.from_dense, dmp_fateman_poly_F_3(2, ZZ))
+    H, cff, cfg = R.dmp_zz_heu_gcd(f, g)
+
+    assert H == h and R.dmp_mul(H, cff) == f \
+                  and R.dmp_mul(H, cfg) == g
+
+    H, cff, cfg = R.dmp_rr_prs_gcd(f, g)
+
+    assert H == h and R.dmp_mul(H, cff) == f \
+                  and R.dmp_mul(H, cfg) == g
+
+    R, x, y, z, u, v = ring("x,y,z,u,v", ZZ)
+
+    f, g, h = map(R.from_dense, dmp_fateman_poly_F_3(4, ZZ))
+    H, cff, cfg = R.dmp_inner_gcd(f, g)
+
+    assert H == h and R.dmp_mul(H, cff) == f \
+                  and R.dmp_mul(H, cfg) == g
+
+    R, x, y = ring("x,y", QQ)
+
+    f = QQ(1,2)*x**2 + x + QQ(1,2)
+    g = QQ(1,2)*x + QQ(1,2)
+
+    h = x + 1
+
+    assert R.dmp_qq_heu_gcd(f, g) == (h, g, QQ(1,2))
+    assert R.dmp_ff_prs_gcd(f, g) == (h, g, QQ(1,2))
+
+    R, x, y = ring("x,y", RR)
+
+    f = 2.1*x*y**2 - 2.2*x*y + 2.1*x
+    g = 1.0*x**3
+
+    assert R.dmp_ff_prs_gcd(f, g) == \
+        (1.0*x, 2.1*y**2 - 2.2*y + 2.1, 1.0*x**2)
+
+
+def test_dup_lcm():
+    R, x = ring("x", ZZ)
+
+    assert R.dup_lcm(2, 6) == 6
+
+    assert R.dup_lcm(2*x**3, 6*x) == 6*x**3
+    assert R.dup_lcm(2*x**3, 3*x) == 6*x**3
+
+    assert R.dup_lcm(x**2 + x, x) == x**2 + x
+    assert R.dup_lcm(x**2 + x, 2*x) == 2*x**2 + 2*x
+    assert R.dup_lcm(x**2 + 2*x, x) == x**2 + 2*x
+    assert R.dup_lcm(2*x**2 + x, x) == 2*x**2 + x
+    assert R.dup_lcm(2*x**2 + x, 2*x) == 4*x**2 + 2*x
+
+
+def test_dmp_lcm():
+    R, x, y = ring("x,y", ZZ)
+
+    assert R.dmp_lcm(2, 6) == 6
+    assert R.dmp_lcm(x, y) == x*y
+
+    assert R.dmp_lcm(2*x**3, 6*x*y**2) == 6*x**3*y**2
+    assert R.dmp_lcm(2*x**3, 3*x*y**2) == 6*x**3*y**2
+
+    assert R.dmp_lcm(x**2*y, x*y**2) == x**2*y**2
+
+    f = 2*x*y**5 - 3*x*y**4 - 2*x*y**3 + 3*x*y**2
+    g = y**5 - 2*y**3 + y
+    h = 2*x*y**7 - 3*x*y**6 - 4*x*y**5 + 6*x*y**4 + 2*x*y**3 - 3*x*y**2
+
+    assert R.dmp_lcm(f, g) == h
+
+    f = x**3 - 3*x**2*y - 9*x*y**2 - 5*y**3
+    g = x**4 + 6*x**3*y + 12*x**2*y**2 + 10*x*y**3 + 3*y**4
+    h = x**5 + x**4*y - 18*x**3*y**2 - 50*x**2*y**3 - 47*x*y**4 - 15*y**5
+
+    assert R.dmp_lcm(f, g) == h
+
+
+def test_dmp_content():
+    R, x,y = ring("x,y", ZZ)
+
+    assert R.dmp_content(-2) == 2
+
+    f, g, F = 3*y**2 + 2*y + 1, 1, 0
+
+    for i in range(0, 5):
+        g *= f
+        F += x**i*g
+
+    assert R.dmp_content(F) == f.drop(x)
+
+    R, x,y,z = ring("x,y,z", ZZ)
+
+    assert R.dmp_content(f_4) == 1
+    assert R.dmp_content(f_5) == 1
+
+    R, x,y,z,t = ring("x,y,z,t", ZZ)
+    assert R.dmp_content(f_6) == 1
+
+
+def test_dmp_primitive():
+    R, x,y = ring("x,y", ZZ)
+
+    assert R.dmp_primitive(0) == (0, 0)
+    assert R.dmp_primitive(1) == (1, 1)
+
+    f, g, F = 3*y**2 + 2*y + 1, 1, 0
+
+    for i in range(0, 5):
+        g *= f
+        F += x**i*g
+
+    assert R.dmp_primitive(F) == (f.drop(x), F / f)
+
+    R, x,y,z = ring("x,y,z", ZZ)
+
+    cont, f = R.dmp_primitive(f_4)
+    assert cont == 1 and f == f_4
+    cont, f = R.dmp_primitive(f_5)
+    assert cont == 1 and f == f_5
+
+    R, x,y,z,t = ring("x,y,z,t", ZZ)
+
+    cont, f = R.dmp_primitive(f_6)
+    assert cont == 1 and f == f_6
+
+
+def test_dup_cancel():
+    R, x = ring("x", ZZ)
+
+    f = 2*x**2 - 2
+    g = x**2 - 2*x + 1
+
+    p = 2*x + 2
+    q = x - 1
+
+    assert R.dup_cancel(f, g) == (p, q)
+    assert R.dup_cancel(f, g, include=False) == (1, 1, p, q)
+
+    f = -x - 2
+    g = 3*x - 4
+
+    F = x + 2
+    G = -3*x + 4
+
+    assert R.dup_cancel(f, g) == (f, g)
+    assert R.dup_cancel(F, G) == (f, g)
+
+    assert R.dup_cancel(0, 0) == (0, 0)
+    assert R.dup_cancel(0, 0, include=False) == (1, 1, 0, 0)
+
+    assert R.dup_cancel(x, 0) == (1, 0)
+    assert R.dup_cancel(x, 0, include=False) == (1, 1, 1, 0)
+
+    assert R.dup_cancel(0, x) == (0, 1)
+    assert R.dup_cancel(0, x, include=False) == (1, 1, 0, 1)
+
+    f = 0
+    g = x
+    one = 1
+
+    assert R.dup_cancel(f, g, include=True) == (f, one)
+
+
+def test_dmp_cancel():
+    R, x, y = ring("x,y", ZZ)
+
+    f = 2*x**2 - 2
+    g = x**2 - 2*x + 1
+
+    p = 2*x + 2
+    q = x - 1
+
+    assert R.dmp_cancel(f, g) == (p, q)
+    assert R.dmp_cancel(f, g, include=False) == (1, 1, p, q)
+
+    assert R.dmp_cancel(0, 0) == (0, 0)
+    assert R.dmp_cancel(0, 0, include=False) == (1, 1, 0, 0)
+
+    assert R.dmp_cancel(y, 0) == (1, 0)
+    assert R.dmp_cancel(y, 0, include=False) == (1, 1, 1, 0)
+
+    assert R.dmp_cancel(0, y) == (0, 1)
+    assert R.dmp_cancel(0, y, include=False) == (1, 1, 0, 1)