Graver bases of shifted numerical semigroups with 3 generators

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.