New classes of examples satisfying the three matrix analog of Gerstenhaber's theorem

TL;DR

This paper explores new classes of examples where the three matrix analog of Gerstenhaber's theorem holds, using commutative algebra techniques to establish results in specific cases and proposing an inductive approach for future generalization.

Contribution

The paper introduces new classes of examples satisfying the three matrix analog of Gerstenhaber's theorem and develops a commutative-algebraic framework for analyzing this problem.

Findings

Proved the three matrix analog of Gerstenhaber's theorem for certain classes of examples.

Translated the problem into a module map statement and verified it in special cases.

Suggested an inductive approach for broader cases.

Abstract

In 1961, Gerstenhaber proved the following theorem: if k is a field and X and Y are commuting dxd matrices with entries in k, then the unital k-algebra generated by these matrices has dimension at most d. The analog of this statement for four or more commuting matrices is false. The three matrix version remains open. We use commutative-algebraic techniques to prove that the three matrix analog of Gerstenhaber's theorem is true for some new classes of examples. In particular, we translate this three commuting matrix statement into an equivalent statement about certain maps between modules, and prove that this commutative-algebraic reformulation is true in special cases. We end with ideas for an inductive approach intended to handle the three matrix analog of Gerstenhaber's theorem more generally.