Polynomial Matrices, Splitting Subspaces and Krylov Subspaces over Finite Fields

TL;DR

This paper explores the enumeration of splitting subspaces and their relation to Krylov and polynomial matrix problems over finite fields, providing explicit formulas under certain conditions.

Contribution

It introduces explicit formulas for counting T-splitting subspaces when T's invariant factors meet specific degree conditions, linking to Krylov and polynomial matrix enumeration problems.

Findings

Derived explicit formulas for -m-dimensional T-splitting subspaces.

Established connections between splitting subspaces, Krylov spaces, and polynomial matrices.

Discussed open problems and conditions for enumeration formulas.

Abstract

Let $T$ be a linear operator on an $\mathbb{F}_q$-vector space $V$ of dimension $n$. For any divisor $m$ of $n$, an $m$-dimensional subspace $W$ of $V$ is $T$-splitting if $$ V =W\oplus TW\oplus \cdots \oplus T^{d-1}W, $$ where $d=n/m$. Let $\sigma(m,d;T)$ denote the number of $m$-dimensional $T$-splitting subspaces. Determining $\sigma(m,d;T)$ for an arbitrary operator $T$ is an open problem. This problem is closely related to another open problem on Krylov spaces. We discuss this connection and give explicit formulae for $\sigma(m,d;T)$ in the case where the invariant factors of $T$ satisfy certain degree conditions. A connection with another enumeration problem on polynomial matrices is also discussed.