Theorem 1 (the bounds): finite groups with $|\mathrm{Sym}^+(C)|\le\max\{d,2d-4\}$ and $|\mathrm{Sym}(C)|\le2d$

Proved

CurveSymmetry.paper_sharp_bounds

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

algebraic-geometrycurve-symmetry-leaneuclidean-geometry

Let $f\in\mathbb R[x,y]$ be irreducible in $\mathbb C[x,y]$ and of total degree $d\ge2$ . Identify $(x,y)$ with $x+iy$ and let $C=\{z\in\mathbb C: f(\operatorname{Re}z,\operatorname{Im}z)=0\}$ be the real zero set of $f$ . Assume that $C$ is infinite and is not a circle: there are no $c\in\mathbb C$ and real $R>0$ with $C=\{z\in\mathbb C:|z-c|=R\}$ . Let $\mathrm{Sym}(C)$ be the group of Euclidean isometries $T$ of $\mathbb C$ with $T(C)=C$ , and $\mathrm{Sym}^+(C)$ its subgroup of maps of the form $z\mapsto az+b$ with $|a|=1$ , the orientation-preserving ones.

Then:

$\mathrm{Sym}(C)$ is finite;
$\mathrm{Sym}^+(C)$ is finite;
$\mathrm{Sym}^+(C)$ is cyclic;
the orders of the two groups satisfy

|\mathrm{Sym}^+(C)|\le\max\{d,\,2d-4\},\qquad|\mathrm{Sym}(C)|\le2d.

These are the finiteness statement and the two bounds of Theorem 1, together with the finiteness and cyclicity assertions of Lemma 3. Sharpness and the equality case are stated separately.

Formalization Note: the curve is a subset of $\mathbb C$ ; $\mathrm{Sym}(C)$ and $\mathrm{Sym}^+(C)$ are subgroups of the group ℂ ≃ᵢ ℂ of isometric bijections of $\mathbb C$ , membership meaning $T(z)\in C\iff z\in C$ for all $z$ ; orders are cardinalities (Nat.card).

Preamble

import Definitions.Def_CurveSymmetry_01_RealLoci
import Definitions.Def_CurveSymmetry_02_FamilyBasics
import Definitions.Def_CurveSymmetry_03_IsometriesAndCharts
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.OperatorNorm
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

Formal statement

theorem CurveSymmetry.paper_sharp_bounds {f : RPoly} (hf : GeometricallyIrreducible f)
    (hd : 2 ≤ f.totalDegree) (hinf : (cartesianLocus f).Infinite)
    (hnc : ¬ ∃ c : ℂ, ∃ R : ℝ, 0 < R ∧ cartesianLocus f = Metric.sphere c R) :
    Finite (isometrySetGroup (cartesianLocus f)) ∧
      Finite (directIsometryGroup (cartesianLocus f)) ∧
      IsCyclic (directIsometryGroup (cartesianLocus f)) ∧
      Nat.card (directIsometryGroup (cartesianLocus f)) ≤ max f.totalDegree (2 * f.totalDegree - 4) ∧
      Nat.card (isometrySetGroup (cartesianLocus f)) ≤ 2 * f.totalDegree := by sorry

Source

C. Perassi, Sharp symmetry bounds for real algebraic curves (note), Theorem 1, 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/PaperBounds.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).

View graph

Theorem 1 (the bounds): finite groups with ∣Sym+(C)∣≤max⁡{d,2d−4}|\mathrm{Sym}^+(C)|\le\max\{d,2d-4\}∣Sym+(C)∣≤max{d,2d−4} and ∣Sym(C)∣≤2d|\mathrm{Sym}(C)|\le2d∣Sym(C)∣≤2d

Theorem 1 (the bounds): finite groups with $|\mathrm{Sym}^+(C)|\le\max\{d,2d-4\}$ and $|\mathrm{Sym}(C)|\le2d$