The differentials $dx/y^3$ , $x\,dx/y^3$ and $dx/y^2$ of the quartic $x^4+y^4=2$ are holomorphic

Proved

CurveSymmetry.quarticHolo_mem

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

curve-symmetry-leanfunction-fieldskummer-theory

Let $K_4=\mathbb C(x)[Y]/(Y^4-(2-x^4))$ , with $y$ the class of $Y$ , so that $x^4+y^4=2$ . This is the function field of the quartic $X^4+Y^4=2$ , written as the cover $y^4=2-x^4$ of the $x$ -line; in the coordinates $X=z$ , $Y=\bar z$ the real locus of that quartic is the curve $\operatorname{Re}(z^4)=1$ of Remark 5. Let $\Omega_{K_4/\mathbb C}$ be the module of Kähler differentials of $K_4$ , with universal derivation $d$ .

A valuation ring of $K_4$ is a subring $\mathcal O\subseteq K_4$ with $\xi\in\mathcal O$ or $\xi^{-1}\in\mathcal O$ for every nonzero $\xi\in K_4$ , and a place of $K_4$ is a valuation ring $\mathcal O\ne K_4$ containing $\mathbb C$ . A differential is regular at $\mathcal O$ if it is a $\mathbb C$ -linear combination of differentials $a\,db$ with $a,b\in\mathcal O$ , and $H(K_4)\subseteq\Omega_{K_4/\mathbb C}$ is the space of holomorphic differentials, those regular at every place of $K_4$ .

Then each of the three differentials

\frac{dx}{y^3},\qquad \frac{x\,dx}{y^3},\qquad \frac{y\,dx}{y^3}=\frac{dx}{y^2}

belongs to $H(K_4)$ , that is, is regular at every place of $K_4$ .

With their linear independence, this shows that $H(K_4)$ has dimension at least three: this is the lower bound behind the genus three of $\operatorname{Re}(z^4)=1$ in Remark 5, which separates it from the $m=2$ family, of genus two.

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_mem (i : Fin 3) : quarticHolo i ∈ holomorphicSpace (KummerField 4 fermatQuartic) := 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)

View graph

The differentials dx/y3dx/y^3dx/y3, x dx/y3x\,dx/y^3xdx/y3 and dx/y2dx/y^2dx/y2 of the quartic x4+y4=2x^4+y^4=2x4+y4=2 are holomorphic

The differentials $dx/y^3$ , $x\,dx/y^3$ and $dx/y^2$ of the quartic $x^4+y^4=2$ are holomorphic