On the variety of linear recurrences and numerical semigroups

TL;DR

This paper explores the relationship between linear recurrences and numerical semigroups, proving the existence of specific recurrences that vanish on the gaps of finitely generated semigroups, revealing new structural insights.

Contribution

It establishes the existence of linear recurrences of order M with solutions that vanish precisely on the gaps of a given numerical semigroup, linking recurrences to semigroup structure.

Findings

Existence of linear recurrences vanishing on semigroup gaps

Characterization of solutions related to numerical semigroup gaps

Connection between recurrence order and semigroup generators

Abstract

In this work, we prove the existence of linear recurrences of order M with a non-trivial solution vanishing exactly on the set of gaps (or a subset) of a numerical semigroup S finitely generated by a1 < a2 <...< aN and M = aN. Keywords: numerical semigroups, linear recurrences, generating function.