| import Mathlib.Algebra.MvPolynomial.Equiv |
| import Mathlib.Tactic |
|
|
| namespace OAI |
|
|
| /-! |
| |
|
|
| 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 |
| |
