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arXiv · 2306.06588

Waring Problem for Matrices over Finite Fields

Krishna Kishore·Adrian Vasiu·Sailun Zhan
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Abstract

We prove that for all integers $k \geq 1$, $q\ge (k-1)^4+ 6k$, and $m \geq 1$, every matrix in $ M_m(\mathbb F_q)$ is a sum of two kth powers: $M_m(\mathbb F_q)=\{A^k+B^k|A,B\in M_m(\mathbb F_q)\}$. 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.

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Krishna Kishore, Adrian Vasiu, Sailun Zhan. 2023-06-11. Waring Problem for Matrices over Finite Fields. https://doi.org/10.1016/j.jpaa.2024.107656

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