Prove2Me

The quintic (x2+y2)(x3−3xy2)−3x2y+y3(x^2+y^2)(x^3-3xy^2)-3x^2y+y^3 equals Re⁡(z3(∣z∣2+i))\operatorname{Re}\bigl(z^3(|z|^2+i)\bigr)

Proved
CurveSymmetry.quinticValue_eq

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

algebraic-geometrycurve-symmetry-leanzariski-topology

Let f∈R[x,y]f\in\mathbb{R}[x,y] be the quintic polynomial f(x,y)=(x2+y2)(x3−3xy2)−3x2y+y3f(x,y)=(x^2+y^2)(x^3-3xy^2)-3x^2y+y^3, and identify R2\mathbb{R}^2 with C\mathbb{C} through z=x+iyz=x+iy.

Then for every z∈Cz\in\mathbb{C},

f(Re⁡z,Im⁡z)=Re⁡(z3(∣z∣2+i)).f(\operatorname{Re}z,\operatorname{Im}z)=\operatorname{Re}\bigl(z^3(|z|^2+i)\bigr).

This is the identity of equation (2) of the note. It writes the quintic in the form Re⁡(zm(∣z∣2+α))\operatorname{Re}\bigl(z^m(|z|^2+\alpha)\bigr) of the family (1), with m=3m=3 and α=i\alpha=i; the note uses this quintic as an irreducible curve with a rotation of order six that changes the sign of its equation.

Formalization Note: The quintic is encoded as a real-valued function of z∈Cz\in\mathbb{C}: the printed Cartesian expression evaluated at x=Re⁡zx=\operatorname{Re}z, y=Im⁡zy=\operatorname{Im}z.

Preamble
import Definitions.Def_CurveSymmetry_01_RealLoci
import Definitions.Def_CurveSymmetry_02_FamilyBasics
import Definitions.Def_CurveSymmetry_03_IsometriesAndCharts
import Definitions.Def_CurveSymmetry_04_ProjectiveClosure
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.Analysis.Calculus.FDeriv.Analytic
import Mathlib.Analysis.Calculus.FDeriv.Equiv
import Mathlib.Analysis.Calculus.FDeriv.Mul
import Mathlib.Analysis.Calculus.FDeriv.Pi
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.Separable
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
import Mathlib.Topology.Algebra.Module.FiniteDimension
import Mathlib.Topology.Compactification.OnePoint.ProjectiveLine

open CurveSymmetry
set_option autoImplicit false
Formal statement
theorem CurveSymmetry.quinticValue_eq (z : ℂ) :
    quinticValue z = (z ^ 3 * ((‖z‖ : ℂ) ^ 2 + Complex.I)).re := by sorry
Source
C. Perassi, Sharp symmetry bounds for real algebraic curves (note), equation (2), p. 1, https://github.com/carlok/curve-symmetry-lean/blob/d99bc17a1c397956c05d7417de50f9beed56580f/sharp_symmetry_bounds.pdf. Lean: https://github.com/carlok/curve-symmetry-lean/blob/d99bc17a1c397956c05d7417de50f9beed56580f/lean/QuinticExample.lean (C. Perassi)
Human review

  • Endorsed by Shuze Chen · Oct 7, 2026

    Confirmed by the moderator at approval.

  • Endorsed by carlok · Oct 7, 2026

    Confirmed by the mission captain (proposal self-audit).

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