| import OAI.AlgebraicGeometry.CommutingDerivations.ParameterBezout |
| import OAI.AlgebraicGeometry.AbhyankarSathaye.Reconstruction |
| import OAI.AlgebraicGeometry.AbhyankarSathaye.Quotient |
| import Mathlib.Data.Fin.VecNotation |
| import Mathlib.Tactic.FinCases |
|
|
| namespace OAI |
|
|
| /-! |
| Explicit relative polynomial coordinates with an invertible parameter. |
|
|
| The coefficient ring is arbitrary, including the zero ring. In the localization |
| application it is the Laurent coefficient ring `ℂ[c,c⁻¹]`. The relative |
| presentation is explicitly isomorphic to a polynomial algebra in three variables. |
| -/ |
| noncomputable section |
| namespace AbhyankarSathaye.CommutingDerivations.ParameterBridge |
| open MvPolynomial |
|
|
| variable (K : Type*) [CommRing K] |
|
|
| -- Original independent variable order: h,u,v,w. |
| def rawH : MvPolynomial (Fin 4) K := X 0 |
| def rawU : MvPolynomial (Fin 4) K := X 1 |
| def rawV : MvPolynomial (Fin 4) K := X 2 |
| def rawW : MvPolynomial (Fin 4) K := X 3 |
| def rawX : MvPolynomial (Fin 4) K := rawU K^3+rawH K*rawV K |
| def rawY : MvPolynomial (Fin 4) K := -rawU K^2+rawH K*rawW K |
| def rawS : MvPolynomial (Fin 4) K := S (rawH K) (rawU K) (rawV K) (rawW K) |
| def relation (c : K) : MvPolynomial (Fin 4) K := |
| rawH K-P (rawX K) (rawY K) (rawS K)-C c |
| abbrev Q (c : K) := MvPolynomial (Fin 4) K ⧸ Ideal.span {relation K c} |
| def qmap (c : K) : MvPolynomial (Fin 4) K →ₐ[K] Q K c := |
| Ideal.Quotient.mkₐ K (Ideal.span {relation K c}) |
| def coeff (c : K) : K →+* Q K c := algebraMap K _ |
| def qH (c : K) : Q K c := qmap K c (rawH K) |
| def qU (c : K) : Q K c := qmap K c (rawU K) |
| def qV (c : K) : Q K c := qmap K c (rawV K) |
| def qW (c : K) : Q K c := qmap K c (rawW K) |
| def qX (c : K) : Q K c := qmap K c (rawX K) |
| def qY (c : K) : Q K c := qmap K c (rawY K) |
| def qS (c : K) : Q K c := qmap K c (rawS K) |
| def qAlpha (c d : K) : Q K c := |
| coeff K c d^2*parameterA (coeff K c c) (qX K c) (qS K c) |
| def qBeta (c d : K) : Q K c := |
| coeff K c d^2*parameterB (coeff K c c) (qX K c) (qY K c) (qS K c) |
| def qT (c d : K) : Q K c := |
| parameter (qAlpha K c d) (qBeta K c d) (qU K c) (qV K c) (qW K c) |
|
|
| theorem quotient_relation (c : K) : |
| qH K c = coeff K c c+P (qX K c) (qY K c) (qS K c) := by |
| have hz : qmap K c (relation K c) = 0 := |
| Ideal.Quotient.eq_zero_iff_mem.mpr (Ideal.subset_span (by simp)) |
| have he : qH K c-P (qX K c) (qY K c) (qS K c)-coeff K c c = 0 := by |
| simpa [relation, qH, qX, qY, qS, coeff] using hz |
| calc |
| _ = (qH K c-P (qX K c) (qY K c) (qS K c)-coeff K c c)+ |
| (coeff K c c+P (qX K c) (qY K c) (qS K c)) := by ring |
| _ = _ := by rw [he, zero_add] |
|
|
| theorem quotient_cusp (c : K) : (qX K c)^2+(qY K c)^3 = qS K c*qH K c := by |
| have he := congrArg (qmap K c) |
| (cusp_universal (rawH K) (rawU K) (rawV K) (rawW K)) |
| simpa [qX, qY, qS, qH, rawX, rawY, rawS, mul_comm] using he |
|
|
| theorem quotient_bezout (c d : K) (hunit : c*d=1) : |
| qAlpha K c d*qH K c+qBeta K c d*qY K c = 1 := by |
| have hu : coeff K c c*coeff K c d=1 := by |
| simpa using congrArg (coeff K c) hunit |
| exact parameter_bezout_unit _ _ _ _ _ _ |
| (quotient_cusp K c) (quotient_relation K c) hu |
|
|
| theorem quotient_recover_s (c d : K) (hunit : c*d=1) : |
| coeff K c d*((qX K c+(qS K c)^3)^2+(qY K c-(qS K c)^2)^3) = qS K c := by |
| have hu : coeff K c c*coeff K c d=1 := by |
| simpa using congrArg (coeff K c) hunit |
| exact parameter_recover_s _ _ _ _ _ _ |
| (quotient_cusp K c) (quotient_relation K c) hu |
|
|
| -- Free independent variable order: X,Y,T. |
| def freeS (d : K) : MvPolynomial (Fin 3) K := C d*(X 0^2+X 1^3) |
| def freeX (d : K) : MvPolynomial (Fin 3) K := X 0-(freeS K d)^3 |
| def freeY (d : K) : MvPolynomial (Fin 3) K := X 1+(freeS K d)^2 |
| def freeH (c d : K) : MvPolynomial (Fin 3) K := |
| C c+P (freeX K d) (freeY K d) (freeS K d) |
| def freeAlpha (c d : K) : MvPolynomial (Fin 3) K := |
| C d^2*parameterA (C c) (freeX K d) (freeS K d) |
| def freeBeta (c d : K) : MvPolynomial (Fin 3) K := |
| C d^2*parameterB (C c) (freeX K d) (freeY K d) (freeS K d) |
| def freeU (c d : K) : MvPolynomial (Fin 3) K := |
| inverseU (freeH K c d) (freeX K d) (freeBeta K c d) (X 2) |
| def freeG (c d : K) : MvPolynomial (Fin 3) K := |
| inverseG (freeX K d) (freeY K d) (freeAlpha K c d) (X 2) |
| def freeW (c d : K) : MvPolynomial (Fin 3) K := |
| inverseW (freeH K c d) (freeY K d) (freeS K d) (freeAlpha K c d) |
| (freeBeta K c d) (freeU K c d) (freeG K c d) |
| def freeV (c d : K) : MvPolynomial (Fin 3) K := |
| inverseV (freeU K c d) (freeG K c d) (freeW K c d) |
|
|
| theorem free_cusp (c d : K) (hunit : c*d=1) : |
| (freeX K d)^2+(freeY K d)^3=freeS K d*freeH K c d := by |
| have hu : (C c : MvPolynomial (Fin 3) K)*C d=1 := by |
| simpa using congrArg (C : K →+* MvPolynomial (Fin 3) K) hunit |
| have hs : C c*freeS K d=X 0^2+X 1^3 := by |
| rw [freeS, ← mul_assoc, hu, one_mul] |
| have he := shift_universal (freeX K d) (freeY K d) (freeS K d) |
| have hx : freeX K d+(freeS K d)^3=X 0 := by unfold freeX; ring |
| have hy : freeY K d-(freeS K d)^2=X 1 := by unfold freeY; ring |
| rw [hx, hy, ← hs] at he |
| calc |
| _ = C c*freeS K d+freeS K d*P (freeX K d) (freeY K d) (freeS K d) := |
| (eq_sub_iff_add_eq.mp he).symm |
| _ = _ := by unfold freeH; ring |
|
|
| theorem free_bezout (c d : K) (hunit : c*d=1) : |
| freeAlpha K c d*freeH K c d+freeBeta K c d*freeY K d=1 := by |
| have hu : (C c : MvPolynomial (Fin 3) K)*C d=1 := by |
| simpa using congrArg (C : K →+* MvPolynomial (Fin 3) K) hunit |
| exact parameter_bezout_unit _ _ _ _ _ _ (free_cusp K c d hunit) rfl hu |
|
|
| theorem free_relations (c d : K) (hunit : c*d=1) : |
| (freeU K c d)^3+freeH K c d*freeV K c d=freeX K d ∧ |
| -(freeU K c d)^2+freeH K c d*freeW K c d=freeY K d ∧ |
| S (freeH K c d) (freeU K c d) (freeV K c d) (freeW K c d)=freeS K d := by |
| exact inverse_relations _ _ _ _ _ _ _ |
| (by simpa [mul_comm] using free_cusp K c d hunit) (free_bezout K c d hunit) |
|
|
| def values (c d : K) : Fin 4 → MvPolynomial (Fin 3) K := |
| ![freeH K c d, freeU K c d, freeV K c d, freeW K c d] |
| def evaluation (c d : K) : MvPolynomial (Fin 4) K →ₐ[K] MvPolynomial (Fin 3) K := |
| aeval (values K c d) |
|
|
| @[simp] theorem evaluation_h (c d : K) : evaluation K c d (rawH K)=freeH K c d := by |
| simp [evaluation, values, rawH] |
| @[simp] theorem evaluation_u (c d : K) : evaluation K c d (rawU K)=freeU K c d := by |
| simp [evaluation, values, rawU] |
| @[simp] theorem evaluation_v (c d : K) : evaluation K c d (rawV K)=freeV K c d := by |
| simp [evaluation, values, rawV] |
| @[simp] theorem evaluation_w (c d : K) : evaluation K c d (rawW K)=freeW K c d := by |
| simp [evaluation, values, rawW] |
| @[simp] theorem evaluation_x (c d : K) (hunit : c*d=1) : |
| evaluation K c d (rawX K)=freeX K d := by |
| simpa [rawX] using (free_relations K c d hunit).1 |
| @[simp] theorem evaluation_y (c d : K) (hunit : c*d=1) : |
| evaluation K c d (rawY K)=freeY K d := by |
| simpa [rawY] using (free_relations K c d hunit).2.1 |
| @[simp] theorem evaluation_s (c d : K) (hunit : c*d=1) : |
| evaluation K c d (rawS K)=freeS K d := by |
| simpa [rawS] using (free_relations K c d hunit).2.2 |
|
|
| theorem evaluation_relation (c d : K) (hunit : c*d=1) : |
| evaluation K c d (relation K c)=0 := by |
| simp only [relation, map_sub, map_P, evaluation_h, evaluation_x K c d hunit, |
| evaluation_y K c d hunit, evaluation_s K c d hunit] |
| simp [evaluation, freeH] |
|
|
| def forward (c d : K) (hunit : c*d=1) : Q K c →ₐ[K] MvPolynomial (Fin 3) K := by |
| have hi : Ideal.span {relation K c} ≤ RingHom.ker (evaluation K c d) := by |
| apply Ideal.span_le.mpr |
| intro z hz |
| simp only [Set.mem_singleton_iff] at hz |
| subst z |
| exact evaluation_relation K c d hunit |
| exact Ideal.Quotient.liftₐ _ (evaluation K c d) (fun z hz => hi hz) |
|
|
| @[simp] theorem forward_qmap (c d : K) (hunit : c*d=1) (f) : |
| forward K c d hunit (qmap K c f)=evaluation K c d f := rfl |
|
|
| def backward (c d : K) : MvPolynomial (Fin 3) K →ₐ[K] Q K c := |
| aeval ![qX K c+(qS K c)^3, qY K c-(qS K c)^2, qT K c d] |
|
|
| @[simp] theorem backward_X2 (c d : K) : |
| backward K c d (X 2)=qT K c d := by simp [backward] |
|
|
| @[simp] theorem backward_freeS (c d : K) (hunit : c*d=1) : |
| backward K c d (freeS K d)=qS K c := by |
| simpa [backward, freeS, coeff] using quotient_recover_s K c d hunit |
| @[simp] theorem backward_freeX (c d : K) (hunit : c*d=1) : |
| backward K c d (freeX K d)=qX K c := by |
| simp only [freeX, map_sub, map_pow, backward_freeS K c d hunit] |
| simp [backward] |
| @[simp] theorem backward_freeY (c d : K) (hunit : c*d=1) : |
| backward K c d (freeY K d)=qY K c := by |
| simp only [freeY, map_add, map_pow, backward_freeS K c d hunit] |
| simp [backward] |
| @[simp] theorem backward_freeH (c d : K) (hunit : c*d=1) : |
| backward K c d (freeH K c d)=qH K c := by |
| simpa [freeH, backward_freeX K c d hunit, backward_freeY K c d hunit, |
| backward_freeS K c d hunit, coeff] using (quotient_relation K c).symm |
| @[simp] theorem backward_freeAlpha (c d : K) (hunit : c*d=1) : |
| backward K c d (freeAlpha K c d)=qAlpha K c d := by |
| simp [freeAlpha, qAlpha, coeff, backward_freeX K c d hunit, |
| backward_freeS K c d hunit] |
| @[simp] theorem backward_freeBeta (c d : K) (hunit : c*d=1) : |
| backward K c d (freeBeta K c d)=qBeta K c d := by |
| simp [freeBeta, qBeta, coeff, backward_freeX K c d hunit, |
| backward_freeY K c d hunit, backward_freeS K c d hunit] |
|
|
| theorem backward_free_original (c d : K) (hunit : c*d=1) : |
| backward K c d (freeU K c d)=qU K c ∧ |
| backward K c d (freeV K c d)=qV K c ∧ |
| backward K c d (freeW K c d)=qW K c := by |
| have hx : (qU K c)^3+qH K c*qV K c=qX K c := by |
| simp [qU, qH, qV, qX, rawX] |
| have hy : -(qU K c)^2+qH K c*qW K c=qY K c := by |
| simp [qU, qH, qW, qY, rawY] |
| have hs : S (qH K c) (qU K c) (qV K c) (qW K c)=qS K c := by |
| simp [qH, qU, qV, qW, qS, rawS] |
| have he := original_reconstruction _ _ _ _ _ _ _ _ _ |
| (quotient_bezout K c d hunit) hx hy hs |
| simpa [freeU, freeV, freeW, freeG, backward_freeH K c d hunit, |
| backward_freeX K c d hunit, backward_freeY K c d hunit, |
| backward_freeS K c d hunit, backward_freeAlpha K c d hunit, |
| backward_freeBeta K c d hunit, backward_X2, qT] using he |
|
|
| theorem forward_backward (c d : K) (hunit : c*d=1) : |
| (forward K c d hunit).comp (backward K c d)=AlgHom.id K _ := by |
| apply MvPolynomial.algHom_ext |
| intro i |
| have ht := inverse_parameter (freeH K c d) (freeX K d) (freeY K d) (freeS K d) |
| (freeAlpha K c d) (freeBeta K c d) (X 2) (free_bezout K c d hunit) |
| fin_cases i |
| · simp [backward, qX, qS, evaluation_x K c d hunit, evaluation_s K c d hunit, |
| freeX] |
| · simp [backward, qY, qS, evaluation_y K c d hunit, evaluation_s K c d hunit, |
| freeY] |
| · simpa [backward, qT, qAlpha, qBeta, coeff, qH, qU, qV, qW, qX, qY, qS, |
| evaluation_x K c d hunit, evaluation_y K c d hunit, evaluation_s K c d hunit, |
| freeAlpha, freeBeta, freeU, freeV, freeW, freeG] using ht |
|
|
| theorem backward_forward (c d : K) (hunit : c*d=1) : |
| (backward K c d).comp (forward K c d hunit)=AlgHom.id K _ := by |
| apply Ideal.Quotient.algHom_ext |
| apply MvPolynomial.algHom_ext |
| intro i |
| obtain ⟨hu, hv, hw⟩ := backward_free_original K c d hunit |
| change backward K c d (forward K c d hunit (qmap K c (X i)))=qmap K c (X i) |
| rw [forward_qmap] |
| fin_cases i |
| · simpa [evaluation, values, qH, rawH] using backward_freeH K c d hunit |
| · simpa [evaluation, values, qU, rawU] using hu |
| · simpa [evaluation, values, qV, rawV] using hv |
| · simpa [evaluation, values, qW, rawW] using hw |
|
|
| /-- The complete explicit relative bridge. For a Laurent coefficient ring, |
| `c` is its Laurent variable and `d` its inverse. Both composites are proved. -/ |
| def equivalence (c d : K) (hunit : c*d=1) : |
| Q K c ≃ₐ[K] MvPolynomial (Fin 3) K := |
| AlgEquiv.ofAlgHom (forward K c d hunit) (backward K c d) |
| (forward_backward K c d hunit) (backward_forward K c d hunit) |
|
|
| end AbhyankarSathaye.CommutingDerivations.ParameterBridge |
|
|
| end |
|
|
| end OAI |
|
|
