Abstract :

Let $m, n$ be positive integers and denote by $F_q$ the finite field with $q$ elements. Let $V$ be a vector space of dimension $mn$ over $F_q$ and $T: V \mapsto V$ be a linear transformation. An $m$-dimensional subspace $W$ of $V$ is said to be $T$-splitting if $V = W \oplus TW_i \oplus \ldots \oplus T^{n-1}W$. Determining the number of $m$-dimensional $T$-splitting subspaces for an arbitrary transformation $T$ is an open problem closely related to many problems in combinatorics and cryptography. I will outline con- nections with a theorem of Philip Hall on conjugacy class size in the general linear group and some results of Wilf et al. on the probability of coprime polynomials over finite fields. I will also discuss a general enumeration problem on matrix polynomials which, if solved, would settle the problem of counting $T$-splitting subspaces.