Irreducible polynomials whose real loci correspond under $z\mapsto az+b$ generate the same ideal after substitution

Proved

CurveSymmetry.similarity_span_eq

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

curve-symmetry-leanfunction-fieldskummer-theory

For a polynomial $R\in\mathbb C[X,Y]$ , let $Z(R)=\{z\in\mathbb C:\ R(z,\bar z)=0\}$ be its real locus in the complex coordinates $X=z$ , $Y=\bar z$ . For a real polynomial $f(x,y)$ rewritten in these coordinates, $Z$ is the zero set of $f$ in $\mathbb R^2=\mathbb C$ .

Let $P,Q\in\mathbb C[X,Y]$ be irreducible polynomials whose real loci $Z(P)$ and $Z(Q)$ are infinite. Let $a,b\in\mathbb C$ with $a\ne0$ , and suppose that the similarity $S(z)=az+b$ maps $Z(Q)$ onto $Z(P)$ , that is, $S(Z(Q))=Z(P)$ . Let $\sigma$ be the automorphism of the $\mathbb C$ -algebra $\mathbb C[X,Y]$ given by $\sigma(R)(X,Y)=R(aX+b,\ \bar aY+\bar b)$ , the substitution of $S$ in these coordinates. Then the ideal generated by $P$ is the image under $\sigma^{-1}$ of the ideal generated by $Q$ :

(P)=\sigma^{-1}\bigl((Q)\bigr)=\Bigl(Q\Bigl(\frac{X-b}{a},\ \frac{Y-\bar b}{\bar a}\Bigr)\Bigr).

Equivalently, $P$ is a nonzero constant multiple of $Q\bigl(\frac{X-b}{a},\frac{Y-\bar b}{\bar a}\bigr)$ .

In the check of Remark 5 this is how a similarity between real loci yields an isomorphism of the coordinate rings $\mathbb C[X,Y]/(P)\cong\mathbb C[X,Y]/(Q)$ , hence of the function fields, to which the invariance of the genus applies.

Formalization Note: $\mathbb C[X,Y]$ is MvPolynomial (Fin 2) ℂ and $Z(R)$ is a subset of $\mathbb C$ ; the right-hand side is the image (Ideal.map) of the ideal $(Q)$ under $\sigma^{-1}$ .

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 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.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.Spectrum.Maximal.Localization
import Mathlib.RingTheory.Valuation.ValuationSubring
import Mathlib.Tactic

open CurveSymmetry
set_option autoImplicit false
open Polynomial

Formal statement

theorem CurveSymmetry.similarity_span_eq {P Q : BPoly} (hP : Irreducible P) (hQ : Irreducible Q)
    (hPinf : (realLocus P).Infinite) (hQinf : (realLocus Q).Infinite) {a b : ℂ} (ha : a ≠ 0)
    (h : (fun z : ℂ => a * z + b) '' realLocus Q = realLocus P) :
    Ideal.span {P} =
      (Ideal.span {Q}).map ((simPullEquiv a b ha).symm : BPoly →+* BPoly) := 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/QuarticComparison.lean (C. Perassi)

View graph

Irreducible polynomials whose real loci correspond under z↦az+bz\mapsto az+bz↦az+b generate the same ideal after substitution

Irreducible polynomials whose real loci correspond under $z\mapsto az+b$ generate the same ideal after substitution