| import OAI.AlgebraicGeometry.AbhyankarSathaye.Identities |
| |
| namespace OAI |
| |
| /-! |
| A relative Bézout identity over an arbitrary commutative ring. |
| The inverse of `c` is specified by the equation `c * d = 1`. |
| -/ |
| namespace AbhyankarSathaye.CommutingDerivations |
| |
| variable {K : Type*} [CommRing K] |
| |
| def parameterA (c x s : K) : K := c + 2*s^2*x + 4*s^5 |
| |
| def parameterB (c x y s : K) : K := |
| (3*s^3 - 3*s*y)*(c + 2*s^2*x) - 4*s^4*y^2 |
| |
| theorem parameter_bezout_certificate (h c s x y : K) : |
| parameterA c x s*h + parameterB c x y s*y - c^2 = |
| (c + 2*s^2*x)*(h - c - P x y s) - |
| 4*s^4*(x^2 + y^3 - s*h) := by |
| unfold parameterA parameterB P |
| ring |
| |
| theorem parameter_bezout_unit (h c d s x y : K) |
| (hcusp : x^2 + y^3 = s*h) (hsection : h = c + P x y s) |
| (hunit : c*d = 1) : |
| (d^2*parameterA c x s)*h + (d^2*parameterB c x y s)*y = 1 := by |
| have hz : h - c - P x y s = 0 := by rw [hsection]; ring |
| have hb : parameterA c x s*h + parameterB c x y s*y = c^2 := by |
| apply sub_eq_zero.mp |
| rw [parameter_bezout_certificate, hz, hcusp, sub_self] |
| ring |
| calc |
| _ = d^2*(parameterA c x s*h + parameterB c x y s*y) := by ring |
| _ = d^2*c^2 := by rw [hb] |
| _ = (c*d)^2 := by ring |
| _ = 1 := by rw [hunit]; ring |
| |
| theorem parameter_shift (h c s x y : K) |
| (hcusp : x^2 + y^3 = s*h) (hsection : h = c + P x y s) : |
| (x + s^3)^2 + (y - s^2)^3 = c*s := by |
| rw [shift_universal, hcusp, hsection] |
| ring |
| |
| theorem parameter_recover_s (h c d s x y : K) |
| (hcusp : x^2 + y^3 = s*h) (hsection : h = c + P x y s) |
| (hunit : c*d = 1) : |
| d*((x + s^3)^2 + (y - s^2)^3) = s := by |
| rw [parameter_shift h c s x y hcusp hsection] |
| calc |
| d*(c*s) = (c*d)*s := by ring |
| _ = s := by rw [hunit, one_mul] |
| |
| section Maps |
| variable {L H : Type*} [CommRing L] [FunLike H K L] [RingHomClass H K L] |
| |
| @[simp] theorem map_parameterA (f : H) (c x s : K) : |
| f (parameterA c x s) = parameterA (f c) (f x) (f s) := by |
| simp [parameterA, map_ofNat] |
| |
| @[simp] theorem map_parameterB (f : H) (c x y s : K) : |
| f (parameterB c x y s) = parameterB (f c) (f x) (f y) (f s) := by |
| simp [parameterB, map_ofNat] |
| |
| end Maps |
| end AbhyankarSathaye.CommutingDerivations |
| |
| end OAI |
| |
