Waring Problem For Triangular Matrix Algebra
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866909158684491776 |
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| author | Kaushik, Rahul Singh, Anupam |
| author_facet | Kaushik, Rahul Singh, Anupam |
| contents | The Matrix Waring problem is if we can write every matrix as a sum of $k$-th powers. Here, we look at the same problem for triangular matrix algebra $T_n(\mathbb{F}_q)$ consisting of upper triangular matrices over a finite field. We prove that for all integers $k, n \geq 1$, there exists a constant $\mathcal C(k, n)$, such that for all $q> \mathcal C(k,n)$, every matrix in $T_n(\mathbb{F}_q)$ is a sum of three $k$-th powers. Moreover, if $-1$ is $k$-th power in $\mathbb{F}_q$, then for all $q>\mathcal C(k,n)$, every matrix in $T_n(\mathbb{F}_q)$ is a sum of two $k$-th powers. We make use of Lang-Weil estimates about the number of solutions of equations over finite fields to achieve the desired results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_09598 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Waring Problem For Triangular Matrix Algebra Kaushik, Rahul Singh, Anupam Group Theory Number Theory Rings and Algebras 11P05, 11G25 The Matrix Waring problem is if we can write every matrix as a sum of $k$-th powers. Here, we look at the same problem for triangular matrix algebra $T_n(\mathbb{F}_q)$ consisting of upper triangular matrices over a finite field. We prove that for all integers $k, n \geq 1$, there exists a constant $\mathcal C(k, n)$, such that for all $q> \mathcal C(k,n)$, every matrix in $T_n(\mathbb{F}_q)$ is a sum of three $k$-th powers. Moreover, if $-1$ is $k$-th power in $\mathbb{F}_q$, then for all $q>\mathcal C(k,n)$, every matrix in $T_n(\mathbb{F}_q)$ is a sum of two $k$-th powers. We make use of Lang-Weil estimates about the number of solutions of equations over finite fields to achieve the desired results. |
| title | Waring Problem For Triangular Matrix Algebra |
| topic | Group Theory Number Theory Rings and Algebras 11P05, 11G25 |
| url | https://arxiv.org/abs/2311.09598 |