Prove2Me

C[t][W]/(W2−hα)\mathbb C[t][W]/(W^2-h_\alpha) embeds onto the integral closure of C[t]\mathbb C[t]; its maximal ideals are the points

Proved
CurveSymmetry.family_quadRing_points

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

commutative-algebracurve-symmetry-leandedekind-domains

Let m≥1m\ge 1 be an integer and α∈C\alpha\in\mathbb C with α≠αˉ\alpha\ne\bar\alpha (that is, α\alpha is not real), and let hα(t)=−t (tm+1) (αtm+αˉ)∈C[t]h_\alpha(t)=-t\,(t^m+1)\,(\alpha t^m+\bar\alpha)\in\mathbb C[t]. Let R=C[t][W]/(W2−hα(t))R=\mathbb C[t][W]/(W^2-h_\alpha(t)) and Lα=C(t)[W]/(W2−hα(t))L_\alpha=\mathbb C(t)[W]/(W^2-h_\alpha(t)), with ww the class of WW in LαL_\alpha, and let φ ⁣:R→Lα\varphi\colon R\to L_\alpha be the C[t]\mathbb C[t]-algebra homomorphism with φ(W)=w\varphi(W)=w. For c,d∈Cc,d\in\mathbb C with d2=hα(c)d^2=h_\alpha(c), let ev⁡c,d ⁣:R→C\operatorname{ev}_{c,d}\colon R\to\mathbb C be the evaluation t↦ct\mapsto c, W↦dW\mapsto d. Then:

  1. φ\varphi is injective;
  2. an element x∈Lαx\in L_\alpha is integral over C[t]\mathbb C[t] if and only if x=φ(y)x=\varphi(y) for some y∈Ry\in R;
  3. an ideal m\mathfrak m of RR is maximal if and only if m=ker⁡ev⁡c,d\mathfrak m=\ker\operatorname{ev}_{c,d} for some c,d∈Cc,d\in\mathbb C with d2=hα(c)d^2=h_\alpha(c).

Parts 1 and 2 say that φ\varphi identifies RR with the integral closure of C[t]\mathbb C[t] in LαL_\alpha:

φ ⁣: C[t][W]/(W2−hα(t)) → ∼  {x∈Lα: x integral over C[t]}.\varphi\colon\ \mathbb C[t][W]/\big(W^2-h_\alpha(t)\big)\ \xrightarrow{\ \sim\ }\ \{x\in L_\alpha:\ x\ \text{integral over}\ \mathbb C[t]\}.

In terms of the double cover (7) from the proof of Lemma 4, whose function field is LαL_\alpha, this describes the normalization of VαV_\alpha over the affine tt-line: its coordinate ring is RR, and its points are the pairs (c,d)(c,d) with d2=hα(c)d^2=h_\alpha(c).

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 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.RingTheory.AdjoinRoot
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 Polynomial
Formal statement
theorem CurveSymmetry.family_quadRing_points {m : ℕ} (hm : 0 < m) {α : ℂ} (ha : α ≠ star α) :
    Function.Injective (quadRingMap (familyH m α)) ∧
      (∀ x : QuadField (familyH m α),
        IsIntegral ℂ[X] x ↔ ∃ y, quadRingMap (familyH m α) y = x) ∧
      ∀ 𝔪 : Ideal (QuadRing (familyH m α)), 𝔪.IsMaximal ↔
        ∃ (c d : ℂ) (hd : d ^ 2 = (familyH m α).eval c),
          𝔪 = RingHom.ker (quadEval (familyH m α) c d hd) := by sorry
Source
C. Perassi, Sharp symmetry bounds for real algebraic curves (note): lemma of the formalization, for proof of Lemma 4, equation (7), p. 3, https://github.com/carlok/curve-symmetry-lean/blob/d99bc17a1c397956c05d7417de50f9beed56580f/sharp_symmetry_bounds.pdf. Lean: https://github.com/carlok/curve-symmetry-lean/blob/d99bc17a1c397956c05d7417de50f9beed56580f/lean/QuadraticRing.lean (C. Perassi)

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