A computing strategy and programs to resolve the Gerstenhaber Problem for commuting triples of matrices

TL;DR

This paper presents a MATLAB-based computational approach to address the long-standing Gerstenhaber Problem by constructing three commuting matrices with a generated subalgebra exceeding the matrix size, challenging previous bounds.

Contribution

The authors develop a MATLAB program that can produce counterexamples to the Gerstenhaber Problem for three commuting matrices, providing new computational evidence for the problem.

Findings

Constructed explicit counterexamples with three commuting matrices.

Demonstrated that the subalgebra dimension can exceed matrix size.

Made MATLAB code publicly available for further research.

Abstract

We describe a MATLAB program that could produce a negative answer to the Gerstenhaber Problem by the construction of three commuting $n \times n$ matrices $A,B,C$ over a field $F$ such that the subalgebra $F[A,B,C]$ they generate has dimension greater than $n$. This problem has remained open for nearly 60 years, following Gerstenhaber's surprising result (Annals Math.) that $\dim F[A,B] \le n$ for any two commuting matrices $A,B$. The property fails for four or more commuting matrices. We also make the MATLAB files freely available.