Prove2Me

The derivative h′(t)h'(t) is nonzero in the function field of w2=h(t)w^2=h(t) when hh has a root

Proved
CurveSymmetry.quad_derivative_ne_zero

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

curve-symmetry-leanfunction-fieldsgenus

Let h∈C[t]h\in\mathbb C[t] be a squarefree polynomial such that W2−hW^2-h is irreducible in C(t)[W]\mathbb C(t)[W], and let K=C(t)[W]/(W2−h)K=\mathbb C(t)[W]/(W^2-h) be the function field of the double cover w2=h(t)w^2=h(t), where tt and ww denote the classes of tt and WW. Let h′h' be the derivative of hh and h′(t)h'(t) its image in KK.

Let c∈Cc\in\mathbb C with h(c)=0h(c)=0. Then

h′(t)≠0in K.h'(t)\ne0\quad\text{in }K.

It allows division by h′(t)h'(t) in the regularity criteria at the points over the roots of hh and, applied to the conjugate cover (whose polynomial vanishes at 00), at the place over t=∞t=\infty of the family; in this way it enters the formal proof of the genus clause of Lemma 4 of the note (the normalization of VαV_\alpha has genus mm) and, through the case m=2m=2, of Remark 5.

Formalization Note: the hypotheses that hh is squarefree and that W2−hW^2-h is irreducible are Fact instances; the root cc enters only through the hypothesis h(c)=0h(c)=0.

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_07_Places
import Definitions.Def_CurveSymmetry_08_Differentials
import Definitions.Def_CurveSymmetry_09_Genus
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.DedekindDomain.Dvr
import Mathlib.RingTheory.DiscreteValuationRing.TFAE
import Mathlib.RingTheory.Etale.Field
import Mathlib.RingTheory.Etale.Kaehler
import Mathlib.RingTheory.Kaehler.Basic
import Mathlib.RingTheory.Kaehler.Polynomial
import Mathlib.RingTheory.LocalRing.MaximalIdeal.Basic
import Mathlib.RingTheory.LocalRing.Module
import Mathlib.RingTheory.Localization.AsSubring
import Mathlib.RingTheory.MvPolynomial.Homogeneous
import Mathlib.RingTheory.MvPolynomial.IrreducibleQuadratic
import Mathlib.RingTheory.Nakayama
import Mathlib.RingTheory.Polynomial.Eisenstein.Criterion
import Mathlib.RingTheory.Polynomial.GaussLemma
import Mathlib.RingTheory.RootsOfUnity.Complex
import Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots
import Mathlib.RingTheory.Valuation.ValuationSubring
import Mathlib.Tactic

open CurveSymmetry
set_option autoImplicit false
open Polynomial
variable (h : ℂ[X]) [Fact (Squarefree h)] [Fact (Irreducible (quadRat h))]
  (c d : ℂ) (hd : d ^ 2 = h.eval c)
Formal statement
theorem CurveSymmetry.quad_derivative_ne_zero (hc : h.eval c = 0) :
    algebraMap ℂ[X] (QuadField h) h.derivative ≠ 0 := 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/HolomorphicDifferentials.lean (C. Perassi)

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