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Periodic behavior in families of numerical and affine semigroups via parametric Presburger arithmetic. (English) Zbl 1508.20094

Summary: 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 \(n\).

MSC:

20M14 Commutative semigroups
20M13 Arithmetic theory of semigroups
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