Graver bases of shifted numerical semigroups with 3 generators

Abstract

A numerical semigroup $M$ is a subset of the non-negative integers that is closed under addition. A factorization of $n\in M$ is an expression of $n$ as a sum of generators of $M$, and the Graver basis of $M$ is a collection $\mathrm{\text{Gr}}(M_t)$ of trades between the generators of $M$ that allows for efficient movement between factorizations. Given positive integers $r_1,\dots ,r_k$, consider the family $M_t=〈t+r_1,\dots ,t+r_k$ of “shifted” numerical semigroups whose generators are obtained by translating $r_1,\dots ,r_k$ by an integer parameter $t$. In this paper, we characterize the Graver basis $\mathrm{\text{Gr}}(M_t)$ of $M_t$ for sufficiently large $t$ in the case $k=3$, in the form of a recursive construction of $\mathrm{\text{Gr}}(M_t)$ from that of smaller values of $t$. As a consequence of our result, the number of trades in $\mathrm{\text{Gr}}(M_t)$, when viewed as a function of $t$, is eventually quasilinear. We also obtain a sharp lower bound on the start of quasilinear behavior.