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arXiv · 1911.09136

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

Tristram Bogart·John Goodrick·Kevin Woods
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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 $n$.

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Tristram Bogart, John Goodrick, Kevin Woods. 2019-11-20. Periodic behavior in families of numerical and affine semigroups via parametric Presburger arithmetic. https://arxiv.org/abs/1911.09136

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