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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