Graver bases of shifted numerical semigroups with 3 generators

Abstract

A numerical semigroup [Formula: see text] is a subset of the non-negative integers that is closed under addition. A factorization of [Formula: see text] is an expression of [Formula: see text] as a sum of generators of [Formula: see text], and the Graver basis of [Formula: see text] is a collection [Formula: see text] of trades between the generators of [Formula: see text] that allows for efficient movement between factorizations. Given positive integers [Formula: see text], consider the family [Formula: see text] of “shifted” numerical semigroups whose generators are obtained by translating [Formula: see text] by an integer parameter [Formula: see text]. In this paper, we characterize the Graver basis [Formula: see text] of [Formula: see text] for sufficiently large [Formula: see text] in the case [Formula: see text], in the form of a recursive construction of [Formula: see text] from that of smaller values of [Formula: see text]. As a consequence of our result, the number of trades in [Formula: see text], when viewed as a function of [Formula: see text], is eventually quasilinear. We also obtain a sharp lower bound on the start of quasilinear behavior.