Article Dans Une Revue Journal of Algebra Année : 2026

On matrices commuting with their Frobenius

Sur les matrices commutant avec leur Frobenius

Résumé

The Frobenius of a matrix \(M\) with coefficients in \(\overline{\mathbb F}_p\) is the matrix \(\sigma(M)\) 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 \(\sigma(M)\), \(\sigma^2(M)\), \(\ldots\) in their Frobenius orbit.

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Dates et versions

hal-05474466 , version 1 (23-01-2026)

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Fabian Gundlach, Béranger Seguin. On matrices commuting with their Frobenius. Journal of Algebra, 2026, 697, pp.63-105. ⟨10.1016/j.jalgebra.2026.02.025⟩. ⟨hal-05474466⟩
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