On matrices commuting with their Frobenius

2025/06/10 by Fabian Gundlach, Gundlach, Fabian, Béranger Seguin +1
Computer Science · Engineering · Mathematics · #14G17 #14M15 #15A27 #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #FOS: Mathematics #Matrix Theory and Algorithms #Number Theory (math.NT) #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.2506.08695

openalex publication_date 2025/06/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The Frobenius of a matrix M with coefficients in \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), σ²(M), … in their Frobenius orbit.

Citations

• Counting two-step nilpotent wildly ramified extensions of function fields
• Lang-Weil Type Estimates in Finite Difference Fields
• Mutually annihilating matrices, and a Cohen--Lenstra series for the nodal singularity
• The Hrushovski-Lang-Weil estimates

Related

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• Bounding the Frobenius norm of a q-deformed commutator
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• A p-adic Perron-Frobenius Theorem
• The Perron-Frobenius theorem: some of its applications