Some thoughts on Gerstenhaber's theorem (Q472434)

The paper under review aims at rekindling the interest in extensions of Gerstenhaber's theorem from two commuting matrices to three commuting matrices. The authors mainly discuss possible answers to the question: Must the (unital) subalgebra \(F[A, B, C]\) generated by matrices \(A, B, C\in M_n(F)\) over a field \(F\) have dimension at most \(n\)? For fields of characteristic zero, it is known to be true for \(n\leq 11\) but fails when \(n\geq 29\). A discussion is made on how the gap for \(12\leq n \leq 28\) can be closed. Other situations for which the extension of Gerstenhaber's theorem fails are also pointed out.