Some algebraic results concerning linear recurrence sequences

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 $P_1,\dots ,P_m$ are any monic polynomials over $F$.

27 pages, some notations changed, results added and others improved, references added

References in corpus (1)

• P-canonical forms and Drazin inverses

Cited by in corpus (1)

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