One reflection · skew coupling — instance data

Definition

HodgeComputerOneReflectionSkew

by benediamond · Nov 5, 2026 · Mathlib 0df444a (Lean v4.33.1)

algebraic-geometryhodge-computerone-reflection-skew

The exact fixed polynomials and pole parameter for “One reflection · skew coupling”, over $\mathbb C$ with coordinates $X_0,X_1,X_2,Y_0,Y_1,Y_2$ :

F=F_0 + X_0^3 X_1 Y_0 Y_1 + X_1^3 X_2 Y_0 Y_1 + \frac17 X_0 X_1 Y_0 Y_1 (Y_0^2-Y_1^2).

q=1,\qquad A=X_{0}^{3} Y_{0}^{3}.

where

\begin{aligned} F_0&=X_0^6+X_1^6+X_2^6+Y_0^6+Y_1^6+Y_2^6 \end{aligned}

Definition code

import Definitions.Def_DiamondCertificate

set_option autoImplicit false
noncomputable section
namespace HodgeComputer.OneReflectionSkew
def n : ℕ := 5
def d : ℕ := 6
def F : HodgeComputer.AmbientPoly n := (((((((((((MvPolynomial.X (R := ℂ) (0 : Fin 6)) ^ (6 : ℕ)) + (((((MvPolynomial.X (R := ℂ) (0 : Fin 6)) ^ (3 : ℕ)) * (MvPolynomial.X (R := ℂ) (1 : Fin 6))) * (MvPolynomial.X (R := ℂ) (3 : Fin 6))) * (MvPolynomial.X (R := ℂ) (4 : Fin 6)))) + (((((MvPolynomial.X (R := ℂ) (0 : Fin 6)) * (MvPolynomial.X (R := ℂ) (1 : Fin 6))) * ((MvPolynomial.X (R := ℂ) (3 : Fin 6)) ^ (3 : ℕ))) * (MvPolynomial.X (R := ℂ) (4 : Fin 6))) * MvPolynomial.C (σ := Fin 6) (1 / 7 : ℂ))) - (((((MvPolynomial.X (R := ℂ) (0 : Fin 6)) * (MvPolynomial.X (R := ℂ) (1 : Fin 6))) * (MvPolynomial.X (R := ℂ) (3 : Fin 6))) * ((MvPolynomial.X (R := ℂ) (4 : Fin 6)) ^ (3 : ℕ))) * MvPolynomial.C (σ := Fin 6) (1 / 7 : ℂ))) + ((MvPolynomial.X (R := ℂ) (1 : Fin 6)) ^ (6 : ℕ))) + (((((MvPolynomial.X (R := ℂ) (1 : Fin 6)) ^ (3 : ℕ)) * (MvPolynomial.X (R := ℂ) (2 : Fin 6))) * (MvPolynomial.X (R := ℂ) (3 : Fin 6))) * (MvPolynomial.X (R := ℂ) (4 : Fin 6)))) + ((MvPolynomial.X (R := ℂ) (2 : Fin 6)) ^ (6 : ℕ))) + ((MvPolynomial.X (R := ℂ) (3 : Fin 6)) ^ (6 : ℕ))) + ((MvPolynomial.X (R := ℂ) (4 : Fin 6)) ^ (6 : ℕ))) + ((MvPolynomial.X (R := ℂ) (5 : Fin 6)) ^ (6 : ℕ)))
def A : HodgeComputer.AmbientPoly n := (((MvPolynomial.X (R := ℂ) (0 : Fin 6)) ^ (3 : ℕ)) * ((MvPolynomial.X (R := ℂ) (3 : Fin 6)) ^ (3 : ℕ)))
def q : ℕ := 1
end HodgeComputer.OneReflectionSkew

Source

https://github.com/hodge-computer/sextics/blob/83fbf5c9a0afe9136ce4f1ed324d085a2c899bd0/data/one-reflection-skew/instance.rec

Read-back

What the Lean code literally says, in plain math · GPT-6 (OpenAI Codex; independent auditor)

The declaration $n$ defines the natural number $5$ .

The declaration $d$ defines the natural number $6$ .

The declaration $F$ defines a polynomial in $R=\mathbb C[z_0,z_1,z_2,z_3,z_4,z_5]$ , where $z_i$ is the variable whose index is $i$ in the six-element index set $\{0,1,2,3,4,5\}$ , by

\begin{aligned}F={}&z_0^6+z_0^3z_1z_3z_4+\tfrac17z_0z_1z_3^3z_4-\tfrac17z_0z_1z_3z_4^3\\&+z_1^6+z_1^3z_2z_3z_4+z_2^6+z_3^6+z_4^6+z_5^6.\end{aligned}

The declaration $A$ defines the polynomial $A=z_0^3z_3^3$ in $\mathbb C[z_0,z_1,z_2,z_3,z_4,z_5]$ , with the same variable indexing $0$ through $5$ .

The declaration $q$ defines the natural number $1$ .

Human review

Endorsed by Shuze Chen · Nov 6, 2026
Confirmed by the moderator at approval.
Endorsed by benediamond · Nov 6, 2026
Confirmed by the mission captain (proposal self-audit).

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