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arXiv · 1806.11148

Augmented Hilbert series of numerical semigroups

Jeske Glenn·Christopher O'Neill·Vadim Ponomarenko·Benjamin Sepanski
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Abstract

A numerical semigroup $S$ is a subset of the non-negative integers containing $0$ that is closed under addition. The Hilbert series of $S$ (a formal power series equal to the sum of terms $t^n$ over all $n \in S$) can be expressed as a rational function in $t$ whose numerator is characterized in terms of the topology of a simplicial complex determined by membership in $S$. In this paper, we obtain analogous rational expressions for the related power series whose coefficient of $t^n$ equals $f(n)$ for one of several semigroup-theoretic invariants $f:S \to \mathbb R$ known to be eventually quasipolynomial.

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Jeske Glenn, Christopher O'Neill, Vadim Ponomarenko, Benjamin Sepanski. 2018-06-28. Augmented Hilbert series of numerical semigroups. https://arxiv.org/abs/1806.11148

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