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Let $\mathbb F$ be a finite field. Denote by $M_n(\mathbb F)$ the set of matrices of order $n$ over $\mathbb F$. For a matrix $A\in M_n(\mathbb F)$ what is the cardinality of $C_{ M_n(\mathbb F)}(A)$, the centralizer of $A$ in $M_n(\mathbb F)$? There are papers about it?

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One messy case is when $A$ is unipotent. A calculation of the centralizer in some of those cases is done in a paper of P. Hall and G. Higman (Proc LMS, 1956) –  Geoff Robinson Aug 19 '12 at 15:50

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