Ap\'ery sets of shifted numerical monoids

TL;DR

This paper studies the Apéry sets of shifted numerical monoids, providing characterizations, efficient algorithms for large shifts, and showing that key invariants become quasipolynomial functions of the shift amount.

Contribution

It introduces a characterization of Apéry sets for shifted monoids, an efficient computation method for large shifts, and proves invariants are eventually quasipolynomial.

Findings

Characterization of Apéry sets for large shifts

Efficient algorithm for computing Apéry sets of shifted monoids

Genus and Frobenius number are eventually quasipolynomial functions of the shift

Abstract

A numerical monoid is an additive submonoid of the non-negative integers. Given a numerical monoid $S$, consider the family of "shifted" monoids $M_n$ obtained by adding $n$ to each generator of $S$. In this paper, we characterize the Ap\'ery set of $M_n$ in terms of the Ap\'ery set of the base monoid $S$ when $n$ is sufficiently large. We give a highly efficient algorithm for computing the Ap\'ery set of $M_n$ in this case, and prove that several numerical monoid invariants, such as the genus and Frobenius number, are eventually quasipolynomial as a function of $n$.