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Augmented Hilbert series of numerical semigroups. (English) Zbl 1442.13013

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 \(S\).
The primary goal of this paper is to obtain analogous rational expressions for various augmented Hilbert series, which they define to be series of the form \[ H_{f}(S;t)=\underset{n\in S}{\sum }f\left( n\right) t^{n} \] where \(f\) is some \(S\)-invariant admitting eventually quasipolynomial behavior. They give two such expressions: (i) when \(f(n)\) counts the number of distinct factorization lengths of \(n\) and (ii) when \(f(n)\) is the maximum or minimum factorization length of \(n\).

MSC:

13A50 Actions of groups on commutative rings; invariant theory
20M14 Commutative semigroups

Software:

numericalsgps
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Full Text: arXiv Link

References:

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