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82da87f | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 | import Mathlib.Algebra.MvPolynomial.Equiv
import Mathlib.Tactic
namespace OAI
/-!
# Explicit polynomial exponential shift
This file proves the exponential substitution used in Section 2 of the source,
without presupposing any locally nilpotent derivation infrastructure. The six
variables are ordered `p, s, u, F, J, w`. All formulas work over any commutative
ring: no characteristic-zero or field hypothesis is required for this step.
-/
namespace StableCoordinate.ExponentialShift
open MvPolynomial
variable (k : Type*) [CommRing k]
abbrev P := MvPolynomial (Fin 6) k
noncomputable def p : P k := X 0
noncomputable def s : P k := X 1
noncomputable def u : P k := X 2
noncomputable def F : P k := X 3
noncomputable def J : P k := X 4
noncomputable def w : P k := X 5
noncomputable def x : P k := s k ^ 2 - u k ^ 2 + p k * F k
noncomputable def H : P k := x k ^ 2 * F k - (1 + 2 * s k * x k) * J k -
p k * J k ^ 2 - u k
noncomputable def t (c : k) : P k := C c * w k
noncomputable def d (c : k) : P k := 2 * u k * t k c - p k * t k c ^ 2
noncomputable def substitution (c : k) : Fin 6 → P k :=
![p k,
s k + p k * x k * d k c,
u k - p k * t k c,
F k - (2 * s k * x k + 1) * d k c - p k * x k ^ 2 * d k c ^ 2,
J k - x k ^ 2 * d k c,
w k]
noncomputable def shift (c : k) : P k →ₐ[k] P k :=
aeval (substitution k c)
@[simp] theorem shift_p (c : k) : shift k c (p k) = p k := by
simp [shift, p, substitution]
@[simp] theorem shift_s (c : k) : shift k c (s k) = s k + p k * x k * d k c := by
simp [shift, s, substitution]
@[simp] theorem shift_u (c : k) : shift k c (u k) = u k - p k * t k c := by
simp [shift, u, substitution]
@[simp] theorem shift_F (c : k) : shift k c (F k) =
F k - (2 * s k * x k + 1) * d k c - p k * x k ^ 2 * d k c ^ 2 := by
simp [shift, F, substitution]
@[simp] theorem shift_J (c : k) : shift k c (J k) = J k - x k ^ 2 * d k c := by
simp [shift, J, substitution]
@[simp] theorem shift_w (c : k) : shift k c (w k) = w k := by
simp [shift, w, substitution]
@[simp] theorem shift_t (c b : k) : shift k c (t k b) = t k b := by
simp [t, shift, w, substitution]
@[simp] theorem t_neg (c : k) : t k (-c) = -t k c := by
simp [t]
private theorem quadratic_invariant {A : Type*} [CommRing A]
(p s u F t : A) :
let x := s ^ 2 - u ^ 2 + p * F
let d := 2 * u * t - p * t ^ 2
(s + p * x * d) ^ 2 - (u - p * t) ^ 2 +
p * (F - (2 * s * x + 1) * d - p * x ^ 2 * d ^ 2) = x := by
dsimp
ring
@[simp] theorem shift_x (c : k) : shift k c (x k) = x k := by
simp only [x, map_add, map_sub, map_pow, map_mul, shift_s, shift_u, shift_p, shift_F]
exact quadratic_invariant (p k) (s k) (u k) (F k) (t k c)
@[simp] theorem shift_d_neg (c : k) : shift k c (d k (-c)) = -d k c := by
simp only [d, map_sub, map_mul, map_ofNat, map_pow, shift_u, shift_t, shift_p, t_neg, map_neg]
ring
private theorem inverse_F {A : Type*} [CommRing A] (p s F x d : A) :
(F - (2 * s * x + 1) * d - p * x ^ 2 * d ^ 2) -
(2 * (s + p * x * d) * x + 1) * -d - p * x ^ 2 * (-d) ^ 2 = F := by
ring
@[simp] theorem shift_comp_neg (c : k) :
(shift k c).comp (shift k (-c)) = AlgHom.id k (P k) := by
apply MvPolynomial.algHom_ext
intro i
fin_cases i
· change shift k c (shift k (-c) (p k)) = p k
simp
· change shift k c (shift k (-c) (s k)) = s k
simp only [shift_s, map_add, map_mul, shift_p, shift_x, shift_d_neg]
ring
· change shift k c (shift k (-c) (u k)) = u k
simp only [shift_u, map_sub, map_mul, shift_p, shift_t, t_neg, map_neg]
ring
· change shift k c (shift k (-c) (F k)) = F k
simp only [shift_F, map_sub, map_add, map_mul, map_ofNat, map_one, map_pow,
shift_s, shift_x, shift_d_neg, shift_p]
exact inverse_F (p k) (s k) (F k) (x k) (d k c)
· change shift k c (shift k (-c) (J k)) = J k
simp only [shift_J, map_sub, map_mul, map_pow, shift_x, shift_d_neg]
ring
· change shift k c (shift k (-c) (w k)) = w k
simp
/-- The explicit exponential substitution, with inverse obtained by negating its parameter. -/
noncomputable def shiftEquiv (c : k) : P k ≃ₐ[k] P k :=
AlgEquiv.ofAlgHom (shift k c) (shift k (-c)) (shift_comp_neg k c)
(by simpa only [neg_neg] using shift_comp_neg k (-c))
@[simp] theorem shiftEquiv_apply (c : k) (a : P k) :
shiftEquiv k c a = shift k c a := rfl
private theorem equation_shift {A : Type*} [CommRing A]
(p s u F J x t d : A) :
x ^ 2 * (F - (2 * s * x + 1) * d - p * x ^ 2 * d ^ 2) -
(1 + 2 * (s + p * x * d) * x) * (J - x ^ 2 * d) -
p * (J - x ^ 2 * d) ^ 2 - (u - p * t) =
(x ^ 2 * F - (1 + 2 * s * x) * J - p * J ^ 2 - u) + p * t := by
ring
/-- The crucial equation transport: the automorphism adds `p * c * w` to `H`. -/
theorem shift_H (c : k) : shift k c (H k) = H k + p k * t k c := by
simp only [H, map_sub, map_mul, map_add, map_pow, map_one, map_ofNat,
shift_x, shift_F, shift_s, shift_J, shift_p, shift_u]
exact equation_shift (p k) (s k) (u k) (F k) (J k) (x k) (t k c) (d k c)
/-- Section 2's `exp(w Δ)` is an actual algebra automorphism fixing `p` and
sending `H` to `H + p*w`. -/
theorem exponential_shift_exists :
∃ e : P k ≃ₐ[k] P k, e (p k) = p k ∧ e (H k) = H k + p k * w k := by
refine ⟨shiftEquiv k 1, ?_, ?_⟩
· simp
· simpa [t] using shift_H k 1
end StableCoordinate.ExponentialShift
end OAI
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