Prove2Me

The differentials of C(t)[W]/(W2−hα)\mathbb C(t)[W]/(W^2-h_\alpha) form a one-dimensional space spanned by dtdt

Proved
CurveSymmetry.family_kaehler_dt

by carlok · Oct 7, 2026 · Mathlib 0df444a (Lean v4.33.1)

curve-symmetry-leanfunction-fieldskahler-differentials

Let m≥1m\ge 1 be an integer and α∈C\alpha\in\mathbb C with α≠αˉ\alpha\ne\bar\alpha (that is, α\alpha is not real), and let hα(t)=−t (tm+1) (αtm+αˉ)∈C[t]h_\alpha(t)=-t\,(t^m+1)\,(\alpha t^m+\bar\alpha)\in\mathbb C[t]. Let Lα=C(t)[W]/(W2−hα(t))L_\alpha=\mathbb C(t)[W]/(W^2-h_\alpha(t)), and let ΩLα/C\Omega_{L_\alpha/\mathbb C} be the module of Kähler differentials of LαL_\alpha over C\mathbb C, with universal derivation dd. Then

dim⁡LαΩLα/C=1andΩLα/C=Lα dt.\dim_{L_\alpha}\Omega_{L_\alpha/\mathbb C}=1\qquad\text{and}\qquad\Omega_{L_\alpha/\mathbb C}=L_\alpha\,dt.

LαL_\alpha is the function field of VαV_\alpha in the form of the double cover (7). In the formalization the genus mm in Lemma 4 is computed from an explicit basis ti dt/wt^i\,dt/w, 0≤i<m0\le i<m, of the holomorphic differentials, and this statement provides the one-dimensional space in which they live; orders of vanishing and the genus are not part of it.

Formalization Note: m>0m>0 and α≠αˉ\alpha\ne\bar\alpha are Fact instances.

Preamble
import Definitions.Def_CurveSymmetry_01_RealLoci
import Definitions.Def_CurveSymmetry_02_FamilyBasics
import Definitions.Def_CurveSymmetry_03_IsometriesAndCharts
import Definitions.Def_CurveSymmetry_05_FamilyFunctionField
import Definitions.Def_CurveSymmetry_06_QuadraticRing
import Definitions.Def_CurveSymmetry_08_Differentials
import Mathlib.Algebra.MvPolynomial.Nilpotent
import Mathlib.Algebra.MvPolynomial.NoZeroDivisors
import Mathlib.Algebra.MvPolynomial.PDeriv
import Mathlib.Algebra.Polynomial.FieldDivision
import Mathlib.Algebra.Polynomial.Reverse
import Mathlib.Algebra.Polynomial.SpecificDegree
import Mathlib.Analysis.Complex.Isometry
import Mathlib.Analysis.Complex.OperatorNorm
import Mathlib.Analysis.Complex.Polynomial.Basic
import Mathlib.Analysis.Normed.Affine.MazurUlam
import Mathlib.Data.Complex.Basic
import Mathlib.FieldTheory.RatFunc.Basic
import Mathlib.FieldTheory.Separable
import Mathlib.LinearAlgebra.Basis.Basic
import Mathlib.LinearAlgebra.FiniteDimensional.Defs
import Mathlib.LinearAlgebra.TensorProduct.Basis
import Mathlib.RingTheory.AdjoinRoot
import Mathlib.RingTheory.Etale.Field
import Mathlib.RingTheory.Etale.Kaehler
import Mathlib.RingTheory.Kaehler.Polynomial
import Mathlib.RingTheory.MvPolynomial.Homogeneous
import Mathlib.RingTheory.MvPolynomial.IrreducibleQuadratic
import Mathlib.RingTheory.Polynomial.Eisenstein.Criterion
import Mathlib.RingTheory.Polynomial.GaussLemma
import Mathlib.RingTheory.RootsOfUnity.Complex
import Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots
import Mathlib.Tactic

open CurveSymmetry
set_option autoImplicit false
open Polynomial TensorProduct
variable {m : ℕ} {α : ℂ} [hm : Fact (0 < m)] [ha : Fact (α ≠ star α)]
Formal statement
theorem CurveSymmetry.family_kaehler_dt :
    Module.finrank (QuadField (familyH m α)) Ω[QuadField (familyH m α)⁄ℂ] = 1 ∧
      Submodule.span (QuadField (familyH m α))
          {KaehlerDifferential.D ℂ (QuadField (familyH m α)) (quadT (familyH m α))} = ⊤ := by sorry
Source
C. Perassi, Sharp symmetry bounds for real algebraic curves (note): lemma of the formalization, for Lemma 4, pp. 3-4, https://github.com/carlok/curve-symmetry-lean/blob/d99bc17a1c397956c05d7417de50f9beed56580f/sharp_symmetry_bounds.pdf. Lean: https://github.com/carlok/curve-symmetry-lean/blob/d99bc17a1c397956c05d7417de50f9beed56580f/lean/QuadraticDifferentials.lean (C. Perassi)

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