{"external_id":{"arxiv":["2506.08695"]},"date_created":"2025-06-11T08:16:39Z","status":"public","abstract":[{"text":"The Frobenius of a matrix $M$ with coefficients in $\\bar{\\mathbb F}_p$ is the\r\nmatrix $\\sigma(M)$ obtained by raising each coefficient to the $p$-th power. We\r\nconsider the question of counting matrices with coefficients in $\\mathbb F_q$\r\nwhich commute with their Frobenius, asymptotically when $q$ is a large power of\r\n$p$. We give answers for matrices of size $2$, for diagonalizable matrices, and\r\nfor matrices whose eigenspaces are defined over $\\mathbb F_p$. Moreover, we\r\nexplain what is needed to solve the case of general matrices. We also solve\r\n(for both diagonalizable and general matrices) the corresponding problem when\r\none counts matrices $M$ commuting with all the matrices $\\sigma(M)$,\r\n$\\sigma^2(M)$, $\\ldots$ in their Frobenius orbit.","lang":"eng"}],"language":[{"iso":"eng"}],"_id":"60175","title":"On matrices commuting with their Frobenius","type":"preprint","citation":{"mla":"Gundlach, Fabian, and Beranger Fabrice Seguin. “On Matrices Commuting with Their Frobenius.” <i>ArXiv:2506.08695</i>, 2025.","bibtex":"@article{Gundlach_Seguin_2025, title={On matrices commuting with their Frobenius}, journal={arXiv:2506.08695}, author={Gundlach, Fabian and Seguin, Beranger Fabrice}, year={2025} }","ieee":"F. Gundlach and B. F. Seguin, “On matrices commuting with their Frobenius,” <i>arXiv:2506.08695</i>. 2025.","chicago":"Gundlach, Fabian, and Beranger Fabrice Seguin. “On Matrices Commuting with Their Frobenius.” <i>ArXiv:2506.08695</i>, 2025.","apa":"Gundlach, F., &#38; Seguin, B. F. (2025). On matrices commuting with their Frobenius. In <i>arXiv:2506.08695</i>.","short":"F. Gundlach, B.F. Seguin, ArXiv:2506.08695 (2025).","ama":"Gundlach F, Seguin BF. On matrices commuting with their Frobenius. <i>arXiv:250608695</i>. Published online 2025."},"date_updated":"2025-06-11T08:17:27Z","publication":"arXiv:2506.08695","author":[{"full_name":"Gundlach, Fabian","last_name":"Gundlach","id":"100450","first_name":"Fabian"},{"last_name":"Seguin","full_name":"Seguin, Beranger Fabrice","id":"102487","first_name":"Beranger Fabrice"}],"year":"2025","user_id":"100450"}