Ap\'ery sets of shifted numerical monoids

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 $n$.