Classification of Boolean Cubic Forms in Ten Variables

2026-06-26

AI Summary

This study addresses the classification of non-zero Boolean cubic forms in ten variables under the action of the general linear group GL(10,2). By combining rank-stratified enumeration, polarization techniques, and explicit orbit computation under the group action, the authors achieve the first complete classification of all alternating trilinear forms over GF(2) in dimension ten, identifying exactly 3,691,560 non-zero orbits. For each orbit, they provide a representative with minimal monomial support, the order of its stabilizer subgroup, and a decomposition according to alternating rank, accompanied by a fast and complete set of GL(10,2)-invariant descriptors. The completeness and correctness of the classification are rigorously verified through dual validation using Burnside’s lemma and the orbit–stabilizer theorem.

Abstract

We classify Boolean cubic forms in ten variables up to GL(10,2)-equivalence. The catalog contains all 3691560 nonzero orbits. For every orbit we provide a representative with small monomial count, the stabilizer order, and the alternating rank together with an explicit decomposition. The classification is obtained by rank-stratified enumeration. We verify completeness by the Burnside orbit count and independently by the orbit--stabilizer identity. We also provide a fast, complete GL(10,2)-invariant. By polarization, this gives the first complete classification of alternating trilinear forms in dimension 10 over GF(2).

Problem

• Boolean cubic forms
• GL(10,2)-equivalence
• classification
• alternating trilinear forms
• orbit enumeration

Innovation

• Boolean cubic forms
• GL(10,2)-equivalence
• rank-stratified enumeration
• alternating trilinear forms
• invariant theory