Polynomial Matrices, Splitting Subspaces and Krylov Subspaces over Finite Fields

2021/05/31 by Aggarwal, Divya, Ram, Samrith
#Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2105.15155

Abstract

Let T be a linear operator on an \mathbbF_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⊕ TW⊕ ⋯ ⊕ T^d-1W, where d=n/m. Let σ(m,d;T) denote the number of m-dimensional T-splitting subspaces. Determining σ(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 σ(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.

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