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

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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, most notably the Betti numbers, but also 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 by vectors whose coordinates are polynomial functions of n, and we prove that in this case the Betti numbers are also eventually quasi-polynomial functions of n.

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Notes

  1. This would appear to be a counterexample to a conjecture of Kerstetter and O’Neill in [10] that the first Betti number of a parametric numerical semigroup is always eventually periodic.

References

  1. Assi, A., García Sánchez, P. A.: Numerical Semigroups and Applications, RSME Springer Series, vol. 1, Springer, Berlin (2016)

  2. Bogart, T., Goodrick, J., Woods, K.: Parametric Presburger arithmetic: logic, combinatorics, and quasi-polynomial behavior. Discrete Anal. 4, 34 (2017)

    MathSciNet  MATH  Google Scholar 

  3. Bresinsky, H.: On prime ideals with generic zero \(x_i = t^{n_1}\). Proc. Am. Math. Soc. 47(2), 329–332 (1975)

    MathSciNet  MATH  Google Scholar 

  4. Bruns, W., Herzog, J.: Semigroup rings and simplicial complexes. J. Pure Appl. Algebra 122(3), 185–208 (1997)

    Article  MathSciNet  Google Scholar 

  5. Chapman, S., Kaplan, N., Lemburg, T., Niles, A., Zlogar, C.: Shifts of generators and delta sets of numerical monoids. Int. J. Algebra Comput. 24(5), 655–669 (2014)

    Article  MathSciNet  Google Scholar 

  6. Conaway, R., Gotti, F., Horton, J., O’Neill, C., Pelayo, R., Williams, M., Wissman, B.: Minimal presentations of shifted numerical monoids. Int. J. Algebra Comput. 28(1), 53–68 (2018)

    Article  MathSciNet  Google Scholar 

  7. Delgado, M., García Sánchez, P. A., Rosales, J. C.: Numerical semigroups problem list, arXiv:1304.6552 (2013)

  8. Eisenbud, D.: The Geometry of Syzygies: A Second Course in Algebraic Geometry and Commutative Algebra, Graduate Texts in Mathematics, vol. 229. Springer, Berlin (2005)

    MATH  Google Scholar 

  9. Herzog, J.: Generators and relations of abelian semigroups and semigroup rings. Manuscr. Math. 3(2), 175–193 (1970)

    Article  MathSciNet  Google Scholar 

  10. Kerstetter, F., O’Neill, C.: On parametrized families of numerical semigroups. Commun. Algebra 48(11), 4698–4717 (2020)

    Article  MathSciNet  Google Scholar 

  11. Marker, D.: Model Theory: An Introduction, Graduate Texts in Mathematics, vol. 217. Springer, Berlin (2002)

  12. Ranjana, M., Joydip, S., Indranath, S.: Betti numbers of Bresinsky’s curves in A4. J. Algebra Appl. 18(8), 1950143 (2019)

    Article  MathSciNet  Google Scholar 

  13. Rosales, J.C., Garcia-Sanchez, P.A.: Numerical Semigroups, Developments in Mathematics, vol. 20. Springer, Berlin (2009)

  14. Şahin, M.: Liftings of a monomial curve. Bull. Austral. Math. Soc. 98(2), 1–9 (2018)

    Article  MathSciNet  Google Scholar 

  15. Shen, B.: The parametric Frobenius problem and parametric exclusion. arXiv:1510.01349 (2015)

  16. Sturmfels, B.: Gröbner Bases and Convex Polytopes, University Lecture Series, vol. 8, American Mathematical Soc. (1996)

  17. Sullivant, S.: Algebraic Statistics, Graduate Studies in Mathematics, vol. 194, American Mathematical Soc. (2018)

  18. Woods, K.: The unreasonable ubiquitousness of quasi-polynomials. Electron. J. Combin. 21(1), 1–44 (2014)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors would like to thank Mauricio Velasco for useful and encouraging discussions of some of the ideas presented here while we were preparing this paper. Tristram Bogart and John Goodrick were respectively supported by internal research Grants INV-2017-51-1453 and INV-2018-50-1424 from the Faculty of Sciences of the Universidad de los Andes during their work on this project.

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Authors and Affiliations

  1. Department of Mathematics, Universidad de los Andes, Carrera 1 No. 18A-12, Bogotá, Colombia

    Tristram Bogart & John Goodrick

  2. Department of Mathematics, Oberlin College, Oberlin, OH, 44974, USA

    Kevin Woods

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  1. Tristram Bogart
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  2. John Goodrick
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  3. Kevin Woods
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Correspondence to John Goodrick.

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Communicated by Pascal Weil.

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Bogart, T., Goodrick, J. & Woods, K. Periodic behavior in families of numerical and affine semigroups via parametric Presburger arithmetic. Semigroup Forum 102, 340–356 (2021). https://doi.org/10.1007/s00233-021-10164-3

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