Prove2Me

The complexification of PP evaluated at (z,zˉ)(z,\bar z) equals P(Re⁡z,Im⁡z)P(\operatorname{Re}z,\operatorname{Im}z)

Proved
CurveSymmetry.eval_complexify

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

curve-symmetry-leanpolynomialsreal-algebraic-geometry

Let P∈C[X,Y]P\in\mathbb C[X,Y] and let PC(X,Y)=P(X+Y2,X−Y2i)P^{\mathbb C}(X,Y)=P\bigl(\tfrac{X+Y}{2},\tfrac{X-Y}{2i}\bigr) be its complexification, which expresses PP in the complex coordinates X=zX=z, Y=zˉY=\bar z.

Then for every z∈Cz\in\mathbb C,

PC(z,zˉ)=P(Re⁡z,Im⁡z).P^{\mathbb C}(z,\bar z)=P(\operatorname{Re}z,\operatorname{Im}z).

Consequently {z∈C:PC(z,zˉ)=0}\{z\in\mathbb C:P^{\mathbb C}(z,\bar z)=0\} is the Cartesian zero set {x+iy:x,y∈R, P(x,y)=0}\{x+iy: x,y\in\mathbb R,\ P(x,y)=0\}. For a real equation ff this identifies the curve CC with the zero set, in the coordinates z,zˉz,\bar z, of its complexification, which is the description of CC used from equation (4) on and in Lemma 3 and Theorems 1 and 2.

Preamble
import Definitions.Def_CurveSymmetry_01_RealLoci
import Definitions.Def_CurveSymmetry_02_FamilyBasics
import Mathlib.Algebra.MvPolynomial.Nilpotent
import Mathlib.Algebra.MvPolynomial.NoZeroDivisors
import Mathlib.Algebra.Polynomial.FieldDivision
import Mathlib.Algebra.Polynomial.Reverse
import Mathlib.Analysis.Complex.Isometry
import Mathlib.Analysis.Complex.Polynomial.Basic
import Mathlib.Analysis.Normed.Affine.MazurUlam
import Mathlib.Data.Complex.Basic
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 MvPolynomial
Formal statement
theorem CurveSymmetry.eval_complexify (P : BPoly) (z : ℂ) :
    eval (fun i : Fin 2 => if i = 0 then z else star z) (complexify P) =
      eval (fun i : Fin 2 => if i = 0 then (z.re : ℂ) else (z.im : ℂ)) P := by sorry
Source
C. Perassi, Sharp symmetry bounds for real algebraic curves (note): lemma of the formalization, for Section 2, equation (4), p. 2, https://github.com/carlok/curve-symmetry-lean/blob/d99bc17a1c397956c05d7417de50f9beed56580f/sharp_symmetry_bounds.pdf. Lean: https://github.com/carlok/curve-symmetry-lean/blob/d99bc17a1c397956c05d7417de50f9beed56580f/lean/CartesianCoordinates.lean (C. Perassi)

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