Waring Problem for Matrices over Finite Fields
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866929275300478976 |
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| author | Kishore, Krishna Vasiu, Adrian Zhan, Sailun |
| author_facet | Kishore, Krishna Vasiu, Adrian Zhan, Sailun |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_06588 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Waring Problem for Matrices over Finite Fields Kishore, Krishna Vasiu, Adrian Zhan, Sailun Number Theory Rings and Algebras 11P05, 11T30 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. |
| title | Waring Problem for Matrices over Finite Fields |
| topic | Number Theory Rings and Algebras 11P05, 11T30 |
| url | https://arxiv.org/abs/2306.06588 |