Waring problem for matrices over finite fields

We prove that for all integers k >= 1, q >= ( k - 1) 4 + 6 k, and m >= 1, every matrix in M m ( F q ) is a sum of two kth powers: M m ( F q ) = {A k + B k |A, B is an element of M m ( F q ) }. 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. (c) 2024 Elsevier B.V. All rights reserved.