Prove2Me

The differentials dx/y3dx/y^3, x dx/y3x\,dx/y^3 and dx/y2dx/y^2 of the quartic x4+y4=2x^4+y^4=2 are linearly independent

Proved
CurveSymmetry.quarticHolo_linearIndependent

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

curve-symmetry-leanfunction-fieldskummer-theory

Let K4=C(x)[Y]/(Y4−(2−x4))K_4=\mathbb C(x)[Y]/(Y^4-(2-x^4)), with yy the class of YY, so that x4+y4=2x^4+y^4=2. This is the function field of the quartic X4+Y4=2X^4+Y^4=2, written as the cover y4=2−x4y^4=2-x^4 of the xx-line; in the coordinates X=zX=z, Y=zˉY=\bar z the real locus of that quartic is the curve Re⁡(z4)=1\operatorname{Re}(z^4)=1 of Remark 5. Let ΩK4/C\Omega_{K_4/\mathbb C} be the module of Kähler differentials of K4K_4, with universal derivation dd.

Then the three differentials dx/y3dx/y^3, x dx/y3x\,dx/y^3 and y dx/y3=dx/y2y\,dx/y^3=dx/y^2 are linearly independent over C\mathbb C in ΩK4/C\Omega_{K_4/\mathbb C}: for c0,c1,c2∈Cc_0,c_1,c_2\in\mathbb C,

c0 dxy3+c1 x dxy3+c2 dxy2=0 ⟹ c0=c1=c2=0.c_0\,\frac{dx}{y^3}+c_1\,\frac{x\,dx}{y^3}+c_2\,\frac{dx}{y^2}=0\ \Longrightarrow\ c_0=c_1=c_2=0.

Since the three differentials are also holomorphic (regular at every place of K4K_4), the space of holomorphic differentials of the quartic has dimension at least three; this is the lower bound behind the genus three of Re⁡(z4)=1\operatorname{Re}(z^4)=1 in Remark 5, against the genus two of the m=2m=2 family.

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 Definitions.Def_CurveSymmetry_09_Genus
import Definitions.Def_CurveSymmetry_10_KummerField
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.Dimension.Finrank
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.Basic
import Mathlib.RingTheory.Kaehler.Polynomial
import Mathlib.RingTheory.LocalRing.MaximalIdeal.Basic
import Mathlib.RingTheory.LocalRing.Module
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
Formal statement
theorem CurveSymmetry.quarticHolo_linearIndependent : LinearIndependent ℂ quarticHolo := by sorry
Source
C. Perassi, Sharp symmetry bounds for real algebraic curves (note): lemma of the formalization, for Remark 5, p. 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/FermatHolomorphic.lean (C. Perassi)

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