# Asymptotic properties in the shifted family of a numerical semigroup with few generators

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## Abstract

Let \(a<b\) be positive integers and for any integer \(k\) consider the semigroup \(H_k= \langle k, a+k, b+k \rangle \). If \(K\) is any field, we study the defining relations of the semigroup ring \(K[H_k]\) and its tangent cone \({\text {gr}}_{\mathfrak {m}}K[H_k]\), for \(k\gg 0\). Recent results in Herzog and Stamate (J Algebra 418:8–28, 2014), Jayanthan and Srinivasan (Proc Am Math Soc 141(12):4199–4208, 2013), and Vu (J Algebra 418:66–90, 2014), show that their Betti numbers are eventually periodic in \(k\). We give a better threshold \(k_{a,b}\) than the one already known for which this happens and we describe how the defining equations are periodically changing. We explicitly find all the shifts \(k> k_{a,b}\) that produce complete intersections, completing a result in Jayanthan and Srinivasan (2013). We write the minimal free resolution of \({\text {gr}}_{\mathfrak {m}}K[H_k]\) and we show that its regularity is a quasilinear function for \(k> k_{a,b}\).

## Keywords

Numerical semigroup ring Tangent cone Periodicity Equations Complete intersection Betti numbers## Notes

### Acknowledgments

We would like to thank Jürgen Herzog for useful conversations and suggestions around this work which developped in parallel with our joint paper [7]. We thank Mihai Cipu for carefully reading the manuscript and for the suggested improvements. We thank an anonymous referee for suggestions that led to an improvement of the bound \(k_{a,b}\). We greatfully acknowledge the use of the computer algebra system SINGULAR [3] for our experiments. The author was supported by a Romanian grant awarded by UEFISCDI, program Human Resources, project number 83/2010, PNII-RU code TE-46/2010: Algebraic modeling of some combinatorial objects and computational applications.

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