Balanced factorisations in some algebras

Abstract

We prove that, in any field of characteristic not two and not three except the five-element field, each element decomposes into a product of four factors whose sum vanishes. We also find all $k,n,q$ such that every $n\times n$ matrix over the $q$-element field decomposes into a product of $k$ commuting matrices whose sum vanishes.

4 pages. A Russian version of this paper is at http://halgebra.math.msu.su/staff/klyachko/papers.htm