On the Waring Problem for Matrices over Finite Fields

Simion Breaz

algebra

On the Waring Problem for Matrices over Finite Fields

arXiv:2505.11805

summary

The paper proves that for any finite field 𝔽_q with q ≠ 2 and any matrix size n satisfying q^n > (k‑1)^4, every n×n matrix over 𝔽_q can be written as a sum of two k‑th power matrices.

Abstract

We prove that if $k$ is a positive integer, then for every finite field $F_{q}$ of cardinality $q \neq = 2$ and for every positive integer $n$ such that $q^{n} > (k - 1)^{4}$ , every $n \times n$ matrix over $F_{q}$ can be expressed as a sum of two $k$ -th powers.

V2: A major improvement over the previous version. We revised some of the computational methods, which allowed us to simplify and correct the proofs, as well as improve the main result

Topics & keywords

#finite fields #matrix theory #waring problem #k-th powers #additive decompositionWaring problemfinite fieldmatrixk-th powersum of powers