Some algebraic results concerning linear recurrence sequences

TL;DR

This paper explores the algebraic structure of sets of linear recurrence sequences over a field, revealing their semiring properties and providing compact descriptions of their factorizations.

Contribution

It introduces two graded semiring structures on the set of linear recurrence sequence spaces and characterizes their factorizations in terms of polynomial products and convolutions.

Findings

The set of linear recurrence sequence spaces forms a graded commutative semiring.

Explicit descriptions of polynomial factorizations for these sequence spaces.

Compact formulas for decomposing recurrence sequence spaces into products and convolutions.

Abstract

We study the set $\mathcal{L}_{F}$ of all $F$-vector spaces $L(P)$ where $P$ is monic and splits over $F$ and $L(Q)$ denotes the set of linear recurrence sequences over $F$ with characteristic polynomial $Q$. We show that $\mathcal{L}_{F}$ can be endowed with two structures of graded commutative semiring. This study allows us to obtain, in compact forms, the polynomial $P,Q\in F[X]$ such that $L(P)=\prod_{i=1}^{m}L(P_{i})$ and $L(Q)=L(P_{1})\ast\cdots\ast L(P_{m})$, where $P_{1}, \ldots, P_{m}$ are any monic polynomials over $F$.