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

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.