The problems of classifying pairs of forms and local algebras with zero cube radical are wild

TL;DR

This paper demonstrates that classifying certain pairs of forms and local algebras with zero cube radical is as complex as classifying pairs of matrices up to similarity, indicating these problems are 'wild' and intractable.

Contribution

It establishes the wildness of classifying pairs of forms and local algebras with zero cube radical, linking these problems to the well-known difficulty of matrix similarity classification.

Findings

Classifying pairs of sesquilinear forms with Hermitian second form is wild.

Classifying pairs of bilinear forms with symmetric or skew-symmetric second forms is wild.

Classifying local algebras with zero cube radical and 2-dimensional square radical is wild.

Abstract

We prove that over an algebraically closed field of characteristic not two the problems of classifying pairs of sesquilinear forms in which the second is Hermitian, pairs of bilinear forms in which the second is symmetric (skew-symmetric), and local algebras with zero cube radical and square radical of dimension 2 are hopeless since each of them reduces to the problem of classifying pairs of n-by-n matrices up to simultaneous similarity.