Waring problem for matrices over finite fields

Abstract

We prove that for all integers k≥1, q≥(k−1)4+6k, and m≥1, every matrix in Mm(Fq) is a sum of two kth powers: Mm(Fq)={Ak+Bk|A,B∈Mm(Fq)}. We further generalize and refine this result in the cases when both B and C can be chosen to be invertible, cyclic, or split semisimple, when k is coprime to p, or when m is sufficiently large. We also give a criterion for the Waring problem in terms of stabilizers.