Periodic behavior in families of numerical and affine semigroups via parametric Presburger arithmetic

Abstract

Let \(f_1(n), \ldots , f_k(n)\) be polynomial functions of n. For fixed \(n\in \mathbb {N}\), let \(S_n\subseteq \mathbb {N}\) be the numerical semigroup generated by \(f_1(n),\ldots ,f_k(n)\). As n varies, we show that many invariants of \(S_n\) are eventually quasi-polynomial in n, most notably the Betti numbers, but also the type, the genus, and the size of the \(\Delta \)-set. The tool we use is expressibility in the logical system of parametric Presburger arithmetic. Generalizing to higher dimensional families of semigroups, we also examine affine semigroups \(S_n\subseteq \mathbb {N}^m\) generated by vectors whose coordinates are polynomial functions of n, and we prove that in this case the Betti numbers are also eventually quasi-polynomial functions of n.