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

TL;DR

This paper demonstrates that many invariants of families of numerical and affine semigroups, generated by polynomial functions, exhibit eventual quasi-polynomial behavior as the parameter varies, using parametric Presburger arithmetic.

Contribution

It introduces a logical framework to prove the eventual quasi-polynomial nature of semigroup invariants in polynomially parametrized families, extending to higher dimensions.

Findings

Frobenius number, type, genus, and Δ-set size are eventually quasi-polynomial in n.

Betti numbers of affine semigroups are eventually quasi-polynomial functions of n.

Results apply to both numerical and higher-dimensional affine semigroups.

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 $n$, such as the Frobenius number, 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 be vectors whose coordinates are polynomial functions of $n$, and we prove similar results; for example, the Betti numbers are eventually quasi-polynomial functions of $n$.