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def transpose_list(input):
'\n INPUT:\n\n - ``input`` -- a list of lists, each list of the same length\n\n OUTPUT:\n\n - ``output`` -- a list of lists such that output[i][j] = input[j][i]\n\n EXAMPLES::\n\n sage: from sage.schemes.hyperelliptic_curves.monsky_washnitzer import transpose_list\... |
def helper_matrix(Q):
"\n Compute the (constant) matrix used to calculate the linear\n combinations of the `d(x^i y^j)` needed to eliminate the\n negative powers of `y` in the cohomology (i.e., in\n :func:`reduce_negative`).\n\n INPUT:\n\n - ``Q`` -- cubic polynomial\n\n EXAMPLES::\n\n ... |
def lift(x):
"\n Try to call ``x.lift()``, presumably from the `p`-adics to `\\ZZ`.\n\n If this fails, it assumes the input is a power series, and tries to\n lift it to a power series over `\\QQ`.\n\n This function is just a very kludgy solution to the problem of\n trying to make the reduction code... |
def reduce_negative(Q, p, coeffs, offset, exact_form=None):
"\n Apply cohomology relations to incorporate negative powers of\n `y` into the `y^0` term.\n\n INPUT:\n\n - ``p`` -- prime\n\n - ``Q`` -- cubic polynomial\n\n - ``coeffs`` -- list of length 3 lists. The\n `i`-th list ``[a, b, c]``... |
def reduce_positive(Q, p, coeffs, offset, exact_form=None):
"\n Apply cohomology relations to incorporate positive powers of\n `y` into the `y^0` term.\n\n INPUT:\n\n - ``Q`` -- cubic polynomial\n\n - ``coeffs`` -- list of length 3 lists. The\n `i`-th list [a, b, c] represents\n `y^{2(i -... |
def reduce_zero(Q, coeffs, offset, exact_form=None):
"\n Apply cohomology relation to incorporate `x^2 y^0` term\n into `x^0 y^0` and `x^1 y^0` terms.\n\n INPUT:\n\n - ``Q`` -- cubic polynomial\n\n - ``coeffs`` -- list of length 3 lists. The\n `i`-th list [a, b, c] represents\n `y^{2(i - ... |
def reduce_all(Q, p, coeffs, offset, compute_exact_form=False):
"\n Apply cohomology relations to reduce all terms to a linear\n combination of `dx/y` and `x dx/y`.\n\n INPUT:\n\n - ``Q`` -- cubic polynomial\n\n - ``coeffs`` -- list of length 3 lists. The\n `i`-th list [a, b, c] represents\n ... |
def frobenius_expansion_by_newton(Q, p, M):
'\n Compute the action of Frobenius on `dx/y` and on\n `x dx/y`, using Newton\'s method (as suggested in Kedlaya\'s\n paper [Ked2001]_).\n\n (This function does *not* yet use the cohomology relations - that\n happens afterwards in the "reduction" step.)\n... |
def frobenius_expansion_by_series(Q, p, M):
'\n Compute the action of Frobenius on `dx/y` and on `x dx/y`, using a\n series expansion.\n\n (This function computes the same thing as\n frobenius_expansion_by_newton(), using a different method.\n Theoretically the Newton method should be asymptoticall... |
def adjusted_prec(p, prec):
"\n Compute how much precision is required in ``matrix_of_frobenius`` to\n get an answer correct to ``prec`` `p`-adic digits.\n\n The issue is that the algorithm used in\n :func:`matrix_of_frobenius` sometimes performs divisions by `p`,\n so precision is lost during the ... |
def matrix_of_frobenius(Q, p, M, trace=None, compute_exact_forms=False):
"\n Compute the matrix of Frobenius on Monsky-Washnitzer cohomology,\n with respect to the basis `(dx/y, x dx/y)`.\n\n INPUT:\n\n - ``Q`` -- cubic polynomial `Q(x) = x^3 + ax + b`\n defining an elliptic curve `E` by\n `... |
def matrix_of_frobenius_hyperelliptic(Q, p=None, prec=None, M=None):
"\n Compute the matrix of Frobenius on Monsky-Washnitzer cohomology,\n with respect to the basis `(dx/2y, x dx/2y, ...x^{d-2} dx/2y)`, where\n `d` is the degree of `Q`.\n\n INPUT:\n\n - ``Q`` -- monic polynomial `Q(x)`\n\n - ``... |
class SpecialHyperellipticQuotientElement(CommutativeAlgebraElement):
"\n Element in the Hyperelliptic quotient ring.\n\n EXAMPLES::\n\n sage: R.<x> = QQ['x']\n sage: E = HyperellipticCurve(x^5 - 36*x + 1)\n sage: x,y = E.monsky_washnitzer_gens()\n sage: MW = x.parent()\n ... |
class SpecialHyperellipticQuotientRing(UniqueRepresentation, CommutativeAlgebra):
'\n The special hyperelliptic quotient ring.\n '
_p = None
def __init__(self, Q, R=None, invert_y=True):
"\n Initialize ``self``.\n\n TESTS::\n\n sage: R.<x> = QQ['x']\n sag... |
class MonskyWashnitzerDifferential(ModuleElement):
"\n An element of the Monsky-Washnitzer ring of differentials, of\n the form `F dx/2y`.\n\n EXAMPLES::\n\n sage: R.<x> = QQ['x']\n sage: C = HyperellipticCurve(x^5 - 4*x + 4)\n sage: x,y = C.monsky_washnitzer_gens()\n sage: MW... |
class MonskyWashnitzerDifferentialRing(UniqueRepresentation, Module):
'\n A ring of Monsky--Washnitzer differentials over ``base_ring``.\n '
def __init__(self, base_ring):
"\n Initialize ``self``.\n\n TESTS::\n\n sage: R.<x> = QQ['x']\n sage: E = Hyperellipti... |
def is_Jacobian(J):
"\n Return True if `J` is of type Jacobian_generic.\n\n EXAMPLES::\n\n sage: from sage.schemes.jacobians.abstract_jacobian import Jacobian, is_Jacobian\n sage: P2.<x, y, z> = ProjectiveSpace(QQ, 2)\n sage: C = Curve(x^3 + y^3 + z^3)\n sage: J = Jacobian(C)\n ... |
def Jacobian(C):
'\n EXAMPLES::\n\n sage: from sage.schemes.jacobians.abstract_jacobian import Jacobian\n sage: P2.<x, y, z> = ProjectiveSpace(QQ, 2)\n sage: C = Curve(x^3 + y^3 + z^3)\n sage: Jacobian(C)\n Jacobian of Projective Plane Curve over Rational Field defined by x^3... |
@richcmp_method
class Jacobian_generic(Scheme):
'\n Base class for Jacobians of projective curves.\n\n The input must be a projective curve over a field.\n\n EXAMPLES::\n\n sage: from sage.schemes.jacobians.abstract_jacobian import Jacobian\n sage: P2.<x, y, z> = ProjectiveSpace(QQ, 2)\n ... |
class ProjectiveConic_field(ProjectivePlaneCurve_field):
"\n Create a projective plane conic curve over a field.\n See ``Conic`` for full documentation.\n\n EXAMPLES::\n\n sage: K = FractionField(PolynomialRing(QQ, 't'))\n sage: P.<X, Y, Z> = K[]\n sage: Conic(X^2 + Y^2 - Z^2)\n ... |
class ProjectiveConic_finite_field(ProjectiveConic_field, ProjectivePlaneCurve_finite_field):
"\n Create a projective plane conic curve over a finite field.\n\n See ``Conic`` for full documentation.\n\n EXAMPLES::\n\n sage: K.<a> = FiniteField(9, 'a')\n sage: P.<X, Y, Z> = K[]\n sage... |
class ProjectiveConic_number_field(ProjectiveConic_field):
"\n Create a projective plane conic curve over a number field.\n See ``Conic`` for full documentation.\n\n EXAMPLES::\n\n sage: K.<a> = NumberField(x^3 - 2, 'a')\n sage: P.<X, Y, Z> = K[]\n sage: Conic(X^2 + Y^2 - a*Z^2)\n ... |
class ProjectiveConic_rational_field(ProjectiveConic_number_field):
'\n Create a projective plane conic curve over `\\QQ`.\n\n See ``Conic`` for full documentation.\n\n EXAMPLES::\n\n sage: P.<X, Y, Z> = QQ[]\n sage: Conic(X^2 + Y^2 - 3*Z^2)\n Projective Conic Curve over Rational Fie... |
class ProjectiveConic_rational_function_field(ProjectiveConic_field):
"\n Create a projective plane conic curve over a rational function field\n `F(t)`, where `F` is any field.\n\n The algorithms used in this class come mostly from [HC2006]_.\n\n EXAMPLES::\n\n sage: K = FractionField(Polynomia... |
def Conic(base_field, F=None, names=None, unique=True):
"\n Return the plane projective conic curve defined by ``F``\n over ``base_field``.\n\n The input form ``Conic(F, names=None)`` is also accepted,\n in which case the fraction field of the base ring of ``F``\n is used as base field.\n\n INPU... |
def QuarticCurve(F, PP=None, check=False):
"\n Return the quartic curve defined by the polynomial ``F``.\n\n INPUT:\n\n - ``F`` -- a polynomial in three variables, homogeneous of degree 4\n\n - ``PP`` -- a projective plane (default: None)\n\n - ``check`` -- whether to check for smoothness or not (d... |
def is_QuarticCurve(C):
"\n Check whether ``C`` is a Quartic Curve.\n\n EXAMPLES::\n\n sage: from sage.schemes.plane_quartics.quartic_generic import is_QuarticCurve\n sage: x,y,z = PolynomialRing(QQ, ['x','y','z']).gens()\n sage: Q = QuarticCurve(x**4 + y**4 + z**4)\n sage: is_Qu... |
class QuarticCurve_generic(projective_curve.ProjectivePlaneCurve):
def _repr_type(self):
"\n Return the representation of ``self``.\n\n EXAMPLES::\n\n sage: x,y,z = PolynomialRing(QQ, ['x','y','z']).gens()\n sage: Q = QuarticCurve(x**4 + y**4 + z**4)\n sage:... |
class SchemeHomset_points_product_projective_spaces_ring(SchemeHomset_points):
"\n Set of rational points of a product of projective spaces.\n\n INPUT: See :class:`~sage.schemes.generic.homset.SchemeHomset_generic`.\n\n EXAMPLES::\n\n sage: from sage.schemes.product_projective.homset import Scheme... |
class SchemeHomset_points_product_projective_spaces_field(SchemeHomset_points_product_projective_spaces_ring):
def points(self, **kwds):
"\n Return some or all rational points of a projective scheme.\n\n Over a finite field, all points are returned. Over an infinite field, all points satisf... |
class ProductProjectiveSpaces_morphism_ring(SchemeMorphism_polynomial):
'\n The class of morphisms on products of projective spaces.\n\n The components are projective space morphisms.\n\n EXAMPLES::\n\n sage: T.<x,y,z,w,u> = ProductProjectiveSpaces([2, 1], QQ)\n sage: H = T.Hom(T)\n ... |
class ProductProjectiveSpaces_point_ring(SchemeMorphism_point):
'\n The class of points on products of projective spaces.\n\n The components are projective space points.\n\n EXAMPLES::\n\n sage: T.<x,y,z,w,u> = ProductProjectiveSpaces([2, 1], QQ)\n sage: T.point([1, 2, 3, 4, 5])\n (1... |
class ProductProjectiveSpaces_point_field(ProductProjectiveSpaces_point_ring):
def intersection_multiplicity(self, X):
'\n Return the intersection multiplicity of the codomain of this point and subscheme ``X`` at this point.\n\n This uses the subscheme implementation of intersection_multipl... |
class ProductProjectiveSpaces_point_finite_field(ProductProjectiveSpaces_point_field):
pass
|
def enum_product_projective_rational_field(X, B):
'\n Enumerate projective, rational points on scheme ``X`` of height up to\n bound ``B``.\n\n INPUT:\n\n - ``X`` -- a scheme or set of abstract rational points of a scheme\n\n - ``B`` -- a positive integer bound\n\n OUTPUT:\n\n - a list contain... |
def enum_product_projective_number_field(X, **kwds):
"\n Enumerates product projective points on scheme ``X`` defined over a number field.\n\n Simply checks all of the points of absolute height of at most ``B``\n and adds those that are on the scheme to the list.\n\n This algorithm computes 2 lists: L... |
def enum_product_projective_finite_field(X):
'\n Enumerates projective points on scheme ``X`` defined over a finite field.\n\n INPUT:\n\n - ``X`` - a scheme defined over a finite field or a set of abstract\n rational points of such a scheme.\n\n OUTPUT:\n\n - a list containing the projective ... |
def sieve(X, bound):
'\n Returns the list of all rational points on scheme\n ``X`` of height up to ``bound``.\n\n ALGORITHM:\n\n Main idea behind the algorithm is to find points modulo primes\n and then reconstruct them using chinese remainder theorem.\n We compute the points modulo primes paral... |
def is_ProductProjectiveSpaces(x):
"\n Return True if ``x`` is a product of projective spaces.\n\n This is an ambient space defined by `\\mathbb{P}^n_R \\times \\cdots \\times \\mathbb{P}^m_R`,\n where `R` is a ring and `n,\\ldots, m\\geq 0` are integers.\n\n OUTPUT: Boolean.\n\n EXAMPLES::\n\n ... |
def ProductProjectiveSpaces(n, R=None, names='x'):
"\n Return the Cartesian product of projective spaces.\n\n The input ``n`` is either a list of projective space over the same base\n ring or the list of dimensions, ``R`` the base ring, and ``names`` the\n variable names.\n\n INPUT:\n\n - ``n`` ... |
class ProductProjectiveSpaces_ring(AmbientSpace):
'\n Cartesian product of projective spaces `\\mathbb{P}^{n_1} \\times \\cdots \\times \\mathbb{P}^{n_r}`.\n\n EXAMPLES::\n\n sage: P.<x0,x1,x2,x3,x4> = ProductProjectiveSpaces([1, 2], QQ); P\n Product of projective spaces P^1 x P^2 over Rationa... |
class ProductProjectiveSpaces_field(ProductProjectiveSpaces_ring):
def _point(self, *args, **kwds):
"\n Construct a point.\n\n For internal use only. See :mod:`morphism` for details.\n\n EXAMPLES::\n\n sage: u = QQ['u'].0\n sage: K = NumberField(u^2 - 2, 'v') ... |
class ProductProjectiveSpaces_finite_field(ProductProjectiveSpaces_field):
def _point(self, *args, **kwds):
'\n Construct a point.\n\n For internal use only. See :mod:`morphism` for details.\n\n EXAMPLES::\n\n sage: P = ProductProjectiveSpaces([1, 2], GF(11))\n ... |
class AlgebraicScheme_subscheme_product_projective(AlgebraicScheme_subscheme_projective):
'\n Construct an algebraic subscheme of a product of projective spaces.\n\n .. WARNING::\n\n You should not create objects of this class directly. The\n preferred method to construct such subschemes is to... |
def ZZ_points_of_bounded_height(PS, dim, bound):
'\n Return an iterator of the points in ``self`` of absolute multiplicative\n height of at most ``bound`` in the rational field.\n\n INPUT:\n\n - ``PS`` -- a projective space\n\n - ``dim`` -- a positive integer\n\n - ``bound`` -- a positive intege... |
def QQ_points_of_bounded_height(PS, dim, bound, normalize=False):
'\n Return an iterator of the points in ``self`` of absolute multiplicative\n height of at most ``bound`` in the rational field.\n\n INPUT:\n\n - ``PS`` -- a projective space\n\n - ``dim`` -- a positive integer\n\n - ``bound`` -- ... |
def IQ_points_of_bounded_height(PS, K, dim, bound):
'\n Return an iterator of the points in ``self`` of absolute multiplicative\n height of at most ``bound`` in the imaginary quadratic field ``K``.\n\n INPUT:\n\n - ``PS`` -- a projective space\n\n - ``K`` -- a number field\n\n - ``dim`` -- a pos... |
def points_of_bounded_height(PS, K, dim, bound, prec=53):
"\n Return an iterator of the points in ``K`` with dimension ``dim`` of\n absolute multiplicative height of at most ``bound``.\n\n ALGORITHM:\n\n This is an implementation of Algorithm 6 in [Krumm2016]_.\n\n INPUT:\n\n - ``PS`` -- a proje... |
class SchemeHomset_points_projective_field(SchemeHomset_points):
'\n Set of rational points of a projective variety over a field.\n\n INPUT:\n\n See :class:`SchemeHomset_generic`.\n\n EXAMPLES::\n\n sage: from sage.schemes.projective.projective_homset import SchemeHomset_points_projective_field... |
class SchemeHomset_points_projective_ring(SchemeHomset_points):
'\n Set of rational points of a projective variety over a commutative ring.\n\n INPUT:\n\n See :class:`SchemeHomset_generic`.\n\n EXAMPLES::\n\n sage: from sage.schemes.projective.projective_homset import SchemeHomset_points_projec... |
class SchemeHomset_polynomial_projective_space(SchemeHomset_generic):
'\n Set of morphisms of a projective space.\n\n EXAMPLES::\n\n sage: P.<x,y,z> = ProjectiveSpace(2, QQ)\n sage: Hom(P, P)\n Set of morphisms\n From: Projective Space of dimension 2 over Rational Field\n ... |
class SchemeHomset_points_abelian_variety_field(SchemeHomset_points_projective_field):
"\n Set of rational points of an Abelian variety.\n\n INPUT:\n\n See :class:`SchemeHomset_generic`.\n\n TESTS:\n\n The bug reported at :trac:`1785` is fixed::\n\n sage: # needs sage.rings.number_field sage... |
class SchemeMorphism_polynomial_projective_space(SchemeMorphism_polynomial):
'\n A morphism of schemes determined by rational functions that define\n what the morphism does on points in the ambient projective space.\n\n EXAMPLES::\n\n sage: R.<x,y> = QQ[]\n sage: P1 = ProjectiveSpace(R)\n ... |
class SchemeMorphism_polynomial_projective_space_field(SchemeMorphism_polynomial_projective_space):
def rational_preimages(self, Q, k=1):
"\n Determine all of the rational `k`-th preimages of ``Q`` by this map.\n\n Given a rational point ``Q`` in the domain of this map, return all the ratio... |
class SchemeMorphism_polynomial_projective_space_finite_field(SchemeMorphism_polynomial_projective_space_field):
def _fast_eval(self, x):
'\n Evaluate projective morphism at point described by x.\n\n EXAMPLES::\n\n sage: P.<x,y,z> = ProjectiveSpace(GF(7), 2)\n sage: H ... |
class SchemeMorphism_polynomial_projective_subscheme_field(SchemeMorphism_polynomial_projective_space_field):
'\n Morphisms from subschemes of projective spaces defined over fields.\n '
def __call__(self, x):
'\n Apply this morphism to the point ``x``.\n\n INPUT:\n\n - ``x`... |
class SchemeMorphism_point_projective_ring(SchemeMorphism_point):
'\n A rational point of projective space over a ring.\n\n INPUT:\n\n - ``X`` -- a homset of a subscheme of an ambient projective space over a ring `K`.\n\n - ``v`` -- a list or tuple of coordinates in `K`.\n\n - ``check`` -- boolean... |
class SchemeMorphism_point_projective_field(SchemeMorphism_point_projective_ring):
'\n A rational point of projective space over a field.\n\n INPUT:\n\n - ``X`` -- a homset of a subscheme of an ambient projective space\n over a field `K`.\n\n - ``v`` -- a list or tuple of coordinates in `K`.\n\... |
class SchemeMorphism_point_projective_finite_field(SchemeMorphism_point_projective_field):
def __hash__(self):
"\n Returns the integer hash of this point.\n\n OUTPUT: Integer.\n\n EXAMPLES::\n\n sage: P.<x,y,z> = ProjectiveSpace(GF(5), 2)\n sage: hash(P(2, 1, 2)... |
class SchemeMorphism_point_abelian_variety_field(AdditiveGroupElement, SchemeMorphism_point_projective_field):
'\n A rational point of an abelian variety over a field.\n\n EXAMPLES::\n\n sage: # needs sage.schemes\n sage: E = EllipticCurve([0,0,1,-1,0])\n sage: origin = E(0)\n sa... |
def enum_projective_rational_field(X, B):
'\n Enumerates projective, rational points on scheme ``X`` of height up to\n bound ``B``.\n\n INPUT:\n\n - ``X`` - a scheme or set of abstract rational points of a scheme.\n\n - ``B`` - a positive integer bound.\n\n OUTPUT:\n\n - a list containing t... |
def enum_projective_number_field(X, **kwds):
"\n Enumerates projective points on scheme ``X`` defined over a number field.\n\n Simply checks all of the points of absolute height of at most ``B``\n and adds those that are on the scheme to the list.\n\n This algorithm computes 2 lists: L containing elem... |
def enum_projective_finite_field(X):
"\n Enumerates projective points on scheme ``X`` defined over a finite field.\n\n INPUT:\n\n - ``X`` - a scheme defined over a finite field or a set of abstract\n rational points of such a scheme.\n\n OUTPUT:\n\n - a list containing the projective points o... |
def sieve(X, bound):
"\n Returns the list of all projective, rational points on scheme ``X`` of\n height up to ``bound``.\n\n Height of a projective point `X = (x_1, x_2,\\dots, x_n)` is given by\n `H_X = \\max(y_1, y_2,\\dots, y_n)`, where the values `y_i`\n are the normalized coordinates such tha... |
def is_ProjectiveSpace(x):
"\n Return True if ``x`` is a projective space.\n\n In other words, if ``x`` is an ambient space `\\mathbb{P}^n_R`,\n where `R` is a ring and `n\\geq 0` is an integer.\n\n EXAMPLES::\n\n sage: from sage.schemes.projective.projective_space import is_ProjectiveSpace\n ... |
def ProjectiveSpace(n, R=None, names=None):
"\n Return projective space of dimension ``n`` over the ring ``R``.\n\n EXAMPLES: The dimension and ring can be given in either order.\n\n ::\n\n sage: ProjectiveSpace(3, QQ)\n Projective Space of dimension 3 over Rational Field\n sage: Pro... |
class ProjectiveSpace_ring(UniqueRepresentation, AmbientSpace):
"\n Projective space of dimension `n` over the ring\n `R`.\n\n EXAMPLES::\n\n sage: X.<x,y,z,w> = ProjectiveSpace(3, QQ)\n sage: X.base_scheme()\n Spectrum of Rational Field\n sage: X.base_ring()\n Rational... |
class ProjectiveSpace_field(ProjectiveSpace_ring):
def _point_homset(self, *args, **kwds):
'\n Construct a point Hom-set.\n\n For internal use only. See :mod:`morphism` for details.\n\n TESTS::\n\n sage: P2.<x,y,z> = ProjectiveSpace(2, GF(3))\n sage: P2._point_h... |
class ProjectiveSpace_finite_field(ProjectiveSpace_field):
def _point(self, *args, **kwds):
'\n Construct a point.\n\n For internal use only. See :mod:`morphism` for details.\n\n TESTS::\n\n sage: P2.<x,y,z> = ProjectiveSpace(2, GF(3))\n sage: point_homset = P2.... |
class ProjectiveSpace_rational_field(ProjectiveSpace_field):
def rational_points(self, bound=0):
'\n Returns the projective points `(x_0:\\cdots:x_n)` over\n `\\QQ` with `|x_i| \\leq` bound.\n\n ALGORITHM:\n\n The very simple algorithm works as follows: every point\n `(... |
class AlgebraicScheme_subscheme_projective(AlgebraicScheme_subscheme):
'\n Construct an algebraic subscheme of projective space.\n\n .. WARNING::\n\n You should not create objects of this class directly. The\n preferred method to construct such subschemes is to use\n :meth:`~sage.scheme... |
class AlgebraicScheme_subscheme_projective_field(AlgebraicScheme_subscheme_projective):
'\n Algebraic subschemes of projective spaces defined over fields.\n '
def _morphism(self, *args, **kwds):
'\n Construct a morphism determined by action on points of ``self``.\n\n For internal ... |
class ChowCycle(FGP_Element):
'\n The elements of the Chow group.\n\n .. WARNING::\n\n Do not construct :class:`ChowCycle` objects manually. Instead,\n use the parent :class:`ChowGroup<ChowGroup_class>` to obtain\n generators or Chow cycles corresponding to cones of the fan.\n\n EXAM... |
class ChowGroupFactory(UniqueFactory):
'\n Factory for :class:`ChowGroup_class`.\n '
def create_key_and_extra_args(self, toric_variety, base_ring=ZZ, check=True):
'\n Create a key that uniquely determines the :class:`ChowGroup_class`.\n\n INPUT:\n\n - ``toric_variety`` -- a... |
class ChowGroup_class(FGP_Module_class, WithEqualityById):
'\n The Chow group of a toric variety.\n\n EXAMPLES::\n\n sage: P2 = toric_varieties.P2()\n sage: from sage.schemes.toric.chow_group import ChowGroup_class\n sage: A = ChowGroup_class(P2,ZZ,True); A\n Chow group of 2-d C... |
class ChowGroup_degree_class(SageObject):
"\n A fixed-degree subgroup of the Chow group of a toric variety.\n\n .. WARNING::\n\n Use\n :meth:`~sage.schemes.toric.chow_group.ChowGroup_class.degree`\n to construct :class:`ChowGroup_degree_class` instances.\n\n EXAMPLES::\n\n sag... |
def is_ChowGroup(x) -> bool:
"\n Return whether ``x`` is a :class:`ChowGroup_class`.\n\n INPUT:\n\n - ``x`` -- anything.\n\n OUTPUT: ``True`` or ``False``.\n\n EXAMPLES::\n\n sage: P2 = toric_varieties.P2()\n sage: A = P2.Chow_group()\n sage: from sage.schemes.toric.chow_group ... |
def is_ChowCycle(x) -> bool:
"\n Return whether ``x`` is a :class:`ChowCycle`.\n\n INPUT:\n\n - ``x`` -- anything.\n\n OUTPUT: ``True`` or ``False``.\n\n EXAMPLES::\n\n sage: P2 = toric_varieties.P2()\n sage: A = P2.Chow_group()\n sage: from sage.schemes.toric.chow_group import... |
def is_ToricDivisor(x):
'\n Test whether ``x`` is a toric divisor.\n\n INPUT:\n\n - ``x`` -- anything.\n\n OUTPUT:\n\n - ``True`` if ``x`` is an instance of :class:`ToricDivisor_generic` and\n ``False`` otherwise.\n\n EXAMPLES::\n\n sage: from sage.schemes.toric.divisor import is_Tor... |
def ToricDivisor(toric_variety, arg=None, ring=None, check=True, reduce=True):
'\n Construct a divisor of ``toric_variety``.\n\n INPUT:\n\n - ``toric_variety`` -- a :class:`toric variety\n <sage.schemes.toric.variety.ToricVariety_field>`;\n\n - ``arg`` -- one of the following description of the t... |
class ToricDivisor_generic(Divisor_generic):
'\n Construct a :class:`(toric Weil) divisor <ToricDivisor_generic>` on the\n given toric variety.\n\n INPUT:\n\n - ``v`` -- a list of tuples (multiplicity, coordinate).\n\n - ``parent`` -- :class:`ToricDivisorGroup`. The parent divisor group.\n\n - `... |
class ToricDivisorGroup(DivisorGroup_generic):
'\n The group of (`\\QQ`-T-Weil) divisors on a toric variety.\n\n EXAMPLES::\n\n sage: P2 = toric_varieties.P2()\n sage: P2.toric_divisor_group()\n Group of toric ZZ-Weil divisors\n on 2-d CPR-Fano toric variety covered by 3 affine p... |
class ToricRationalDivisorClassGroup(FreeModule_ambient_field, UniqueRepresentation):
'\n The rational divisor class group of a toric variety.\n\n The **T-Weil divisor class group** `\\mathop{Cl}(X)` of a toric\n variety `X` is a finitely generated abelian group and can contain\n torsion. Its rank equ... |
class ToricRationalDivisorClassGroup_basis_lattice(FreeModule_ambient_pid):
'\n Construct the basis lattice of the ``group``.\n\n INPUT:\n\n - ``group`` -- :class:`toric rational divisor class group\n <ToricRationalDivisorClassGroup>`.\n\n OUTPUT:\n\n - the basis lattice of ``group``.\n\n E... |
def is_CPRFanoToricVariety(x):
'\n Check if ``x`` is a CPR-Fano toric variety.\n\n INPUT:\n\n - ``x`` -- anything.\n\n OUTPUT:\n\n - ``True`` if ``x`` is a :class:`CPR-Fano toric variety\n <CPRFanoToricVariety_field>` and ``False`` otherwise.\n\n .. NOTE::\n\n While projective spaces... |
def CPRFanoToricVariety(Delta=None, Delta_polar=None, coordinate_points=None, charts=None, coordinate_names=None, names=None, coordinate_name_indices=None, make_simplicial=False, base_ring=None, base_field=None, check=True):
'\n Construct a CPR-Fano toric variety.\n\n .. NOTE::\n\n See documentation ... |
class CPRFanoToricVariety_field(ToricVariety_field):
'\n Construct a CPR-Fano toric variety associated to a reflexive polytope.\n\n .. WARNING::\n\n This class does not perform any checks of correctness of input and it\n does assume that the internal structure of the given parameters is\n ... |
class AnticanonicalHypersurface(AlgebraicScheme_subscheme_toric):
'\n Construct an anticanonical hypersurface of a CPR-Fano toric variety.\n\n INPUT:\n\n - ``P_Delta`` -- :class:`CPR-Fano toric variety\n <CPRFanoToricVariety_field>` associated to a reflexive polytope\n `\\Delta`;\n\n - see ... |
class NefCompleteIntersection(AlgebraicScheme_subscheme_toric):
'\n Construct a nef complete intersection in a CPR-Fano toric variety.\n\n INPUT:\n\n - ``P_Delta`` -- a :class:`CPR-Fano toric variety\n <CPRFanoToricVariety_field>` associated to a reflexive polytope\n `\\Delta`;\n\n - see :me... |
def add_variables(field, variables):
'\n Extend ``field`` to include all ``variables``.\n\n INPUT:\n\n - ``field`` - a field;\n\n - ``variables`` - a list of strings.\n\n OUTPUT:\n\n - a fraction field extending the original ``field``, which has all\n ``variables`` among its generators.\n\n... |
class SchemeHomset_toric_variety(SchemeHomset_generic):
"\n Set of homomorphisms between two toric varieties.\n\n EXAMPLES::\n\n sage: P1xP1 = toric_varieties.P1xP1()\n sage: P1 = toric_varieties.P1()\n sage: hom_set = P1xP1.Hom(P1); hom_set\n Set of morphisms\n From: 2... |
class SchemeHomset_points_toric_base(SchemeHomset_points):
'\n Base class for homsets with toric ambient spaces.\n\n INPUT:\n\n - same as for :class:`SchemeHomset_points`.\n\n OUTPUT: A scheme morphism of type :class:`SchemeHomset_points_toric_base`.\n\n EXAMPLES::\n\n sage: P1xP1 = toric_va... |
class SchemeHomset_points_toric_field(SchemeHomset_points_toric_base):
'\n Set of rational points of a toric variety.\n\n You should not use this class directly. Instead, use the\n :meth:`~sage.schemes.generic.scheme.Scheme.point_set` method to\n construct the point set of a toric variety.\n\n INPU... |
class SchemeHomset_points_subscheme_toric_field(SchemeHomset_points_toric_base):
def _enumerator(self):
'\n Return the most suitable enumerator for points.\n\n OUTPUT:\n\n An iterable that yields the coordinates of all points as\n tuples.\n\n EXAMPLES::\n\n s... |
class ToricIdeal(MPolynomialIdeal):
"\n This class represents a toric ideal defined by an integral matrix.\n\n INPUT:\n\n - ``A`` -- integer matrix. The defining matrix of the toric ideal.\n\n - ``names`` -- string (optional). Names for the variables. By\n default, this is ``'z'`` and the variabl... |
class ToricVarietyFactory(SageObject):
'\n The methods of this class construct toric varieties.\n\n .. WARNING::\n\n You need not create instances of this class. Use the\n already-provided object ``toric_varieties`` instead.\n '
_check = True
def _make_ToricVariety(self, name, coor... |
class SchemeMorphism_point_toric_field(SchemeMorphism_point, Morphism):
'\n A point of a toric variety determined by homogeneous coordinates\n in a field.\n\n .. WARNING::\n\n You should not create objects of this class directly. Use the\n :meth:`~sage.schemes.generic.scheme.hom` method of\... |
class SchemeMorphism_polynomial_toric_variety(SchemeMorphism_polynomial, Morphism):
'\n A morphism determined by homogeneous polynomials.\n\n .. WARNING::\n\n You should not create objects of this class directly. Use the\n :meth:`~sage.schemes.generic.scheme.hom` method of\n :class:`tor... |
class SchemeMorphism_orbit_closure_toric_variety(SchemeMorphism, Morphism):
'\n The embedding of an orbit closure.\n\n INPUT:\n\n - ``parent`` -- the parent homset.\n\n - ``defining_cone`` -- the defining cone.\n\n - ``ray_map`` -- a dictionary ``{ambient ray generator: orbit ray\n generator}`... |
class SchemeMorphism_fan_toric_variety(SchemeMorphism, Morphism):
"\n Construct a morphism determined by a fan morphism\n\n .. WARNING::\n\n You should not create objects of this class directly. Use the\n :meth:`~sage.schemes.generic.scheme.hom` method of\n :class:`toric varieties\n ... |
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