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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2506.08695 |
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| _version_ | 1866910045498769408 |
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| author | Gundlach, Fabian Seguin, Béranger |
| author_facet | Gundlach, Fabian Seguin, Béranger |
| contents | The Frobenius of a matrix $M$ with coefficients in $\bar{\mathbb F}_p$ is the matrix $σ(M)$ obtained by raising each coefficient to the $p$-th power. We consider the question of counting matrices with coefficients in $\mathbb F_q$ which commute with their Frobenius, asymptotically when $q$ is a large power of $p$. We give answers for matrices of size $2$, for diagonalizable matrices, and for matrices whose eigenspaces are defined over $\mathbb F_p$. Moreover, we explain what is needed to solve the case of general matrices. We also solve (for both diagonalizable and general matrices) the corresponding problem when one counts matrices $M$ commuting with all the matrices $σ(M)$, $σ^2(M)$, $\ldots$ in their Frobenius orbit. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_08695 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On matrices commuting with their Frobenius Gundlach, Fabian Seguin, Béranger Algebraic Geometry Number Theory 14G17, 15A27, 14M15 The Frobenius of a matrix $M$ with coefficients in $\bar{\mathbb F}_p$ is the matrix $σ(M)$ obtained by raising each coefficient to the $p$-th power. We consider the question of counting matrices with coefficients in $\mathbb F_q$ which commute with their Frobenius, asymptotically when $q$ is a large power of $p$. We give answers for matrices of size $2$, for diagonalizable matrices, and for matrices whose eigenspaces are defined over $\mathbb F_p$. Moreover, we explain what is needed to solve the case of general matrices. We also solve (for both diagonalizable and general matrices) the corresponding problem when one counts matrices $M$ commuting with all the matrices $σ(M)$, $σ^2(M)$, $\ldots$ in their Frobenius orbit. |
| title | On matrices commuting with their Frobenius |
| topic | Algebraic Geometry Number Theory 14G17, 15A27, 14M15 |
| url | https://arxiv.org/abs/2506.08695 |