Semigroup Forum

, Volume 93, Issue 2, pp 225–246 | Cite as

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

  • Dumitru I. Stamate


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 \(k> k_{a,b}\).


Numerical semigroup ring Tangent cone Periodicity Equations Complete intersection Betti numbers 



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.


  1. 1.
    Ramírez Alfonsín, J.L.: The Diophantine Frobenius Problem. Oxford Lecture Series in Mathematics and its Applications. Oxford University Press, Oxford (2005)Google Scholar
  2. 2.
    Bruns, W., Herzog, J.: Cohen–Macaulay Rings, Revised ed. Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (1998)MATHGoogle Scholar
  3. 3.
    Decker, W., Greuel, G.-M., Pfister, G., Schönemann, H.: Singular 3–1-6—A computer algebra system for polynomial computations. (2012)
  4. 4.
    Delorme, C.: Sous-monoïdes d’intersection complète de N. Ann. Sci. Ecole Norm. Sup. (4) 9(1), 145–154 (1976)MathSciNetMATHGoogle Scholar
  5. 5.
    Herzog, J.: Generators and relations of Abelian semigroups and semigroup rings. Manuscr. Math. 3, 175–193 (1970)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Herzog, J.: When is a regular sequence super regular? Nagoya Math. J. 83, 183–195 (1981)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Herzog, J., Stamate, D.I.: On the defining equations of the tangent cone of a numerical semigroup ring. J. Algebra 418, 8–28 (2014)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Jayanthan, A.V., Srinivasan, H.: Periodic occurence of complete intersection monomial curves. Proc. Am. Math. Soc. 141(12), 4199–4208 (2013)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Ohsugi, H., Hibi, T.: Indispensable binomials of finite graphs. J. Algebra Appl. 4(4), 421–434 (2005)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Robbiano, L., Valla, G.: On the equations defining tangent cones. Math. Proc. Camb. Philos. Soc. 88, 281–297 (1980)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Vu, T.: Periodicity of Betti numbers of monomial curves. J. Algebra 418, 66–90 (2014)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Faculty of Mathematics and Computer ScienceUniversity of BucharestBucharestRomania

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