Graver bases of shifted numerical semigroups with 3 generators

TL;DR

This paper characterizes the Graver basis of shifted numerical semigroups with three generators for large shifts, revealing a recursive structure and eventual quasilinear growth in the number of trades.

Contribution

It provides a recursive characterization of Graver bases for three-generator shifted semigroups and establishes the quasilinear growth pattern of the number of trades.

Findings

Recursive construction of Graver bases for large t

Number of trades grows eventually quasilinearly

Sharp lower bound on the onset of quasilinear behavior

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 $Gr(M_t)$ of trades between the generators of $M$ that allows for efficient movement between factorizations. Given positive integers $r_1, \ldots, r_k$, consider the family $M_t = \langle t + r_1, \ldots, t + r_k\rangle$ of "shifted" numerical semigroups whose generators are obtained by translating $r_1, \ldots, r_k$ by an integer parameter $t$. In this paper, we characterize the Graver basis $Gr(M_t)$ of $M_t$ for sufficiently large $t$ in the case $k = 3$, in the form of a recursive construction of $Gr(M_t)$ from that of smaller values of $t$. As a consequence of our result, the number of trades in $Gr(M_t)$, when viewed as a function of $t$, is eventually quasilinear. We also obtain a sharp lower bound on the start of quasilinear behavior.