{
  "id": "2306.06588",
  "version": "v1",
  "published": "2023-06-11T04:43:35.000Z",
  "updated": "2023-06-11T04:43:35.000Z",
  "title": "Waring Problem for Matrices over Finite Fields",
  "authors": [
    "Krishna Kishore",
    "Adrian Vasiu",
    "Sailun Zhan"
  ],
  "comment": "33 pages. Comments are very welcome",
  "categories": [
    "math.NT",
    "math.RA"
  ],
  "abstract": "Let $k$ be a natural number. The Waring problem for square matrices over finite fields, solved by Kishore and Singh, asked for the existence of a constant $C(k)$ such that for all finite fields $\\mathbb F_q$ with $q\\ge C(k)$, each matrix $A\\in M_n(\\mathbb F_q)$ with $n\\ge 2$ is a sum $A=B^k+C^k$ of two $k$-th powers. We present a new proof of this result with the explicit bound $C(k)=(k-1)^4+6k$ and we generalize and refine this result in many cases, including those when $B$ and $C$ are required to be (invertible) (split) semisimple or invertible cyclic, when $k$ is assumed to be coprime to $p$ (the improved bound being $k^3-3k^2+3k$), or when $n$ is big enough.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-06-11T04:43:35.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "finite fields",
      "waring problem",
      "natural number",
      "th powers",
      "explicit bound"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 33,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}