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
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
Assi, A., García Sánchez, P. A.: Numerical Semigroups and Applications, RSME Springer Series, vol. 1, Springer, Berlin (2016)
Bogart, T., Goodrick, J., Woods, K.: Parametric Presburger arithmetic: logic, combinatorics, and quasi-polynomial behavior. Discrete Anal. 4, 34 (2017)
Bresinsky, H.: On prime ideals with generic zero \(x_i = t^{n_1}\). Proc. Am. Math. Soc. 47(2), 329–332 (1975)
Bruns, W., Herzog, J.: Semigroup rings and simplicial complexes. J. Pure Appl. Algebra 122(3), 185–208 (1997)
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)
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)
Delgado, M., García Sánchez, P. A., Rosales, J. C.: Numerical semigroups problem list, arXiv:1304.6552 (2013)
Eisenbud, D.: The Geometry of Syzygies: A Second Course in Algebraic Geometry and Commutative Algebra, Graduate Texts in Mathematics, vol. 229. Springer, Berlin (2005)
Herzog, J.: Generators and relations of abelian semigroups and semigroup rings. Manuscr. Math. 3(2), 175–193 (1970)
Kerstetter, F., O’Neill, C.: On parametrized families of numerical semigroups. Commun. Algebra 48(11), 4698–4717 (2020)
Marker, D.: Model Theory: An Introduction, Graduate Texts in Mathematics, vol. 217. Springer, Berlin (2002)
Ranjana, M., Joydip, S., Indranath, S.: Betti numbers of Bresinsky’s curves in A4. J. Algebra Appl. 18(8), 1950143 (2019)
Rosales, J.C., Garcia-Sanchez, P.A.: Numerical Semigroups, Developments in Mathematics, vol. 20. Springer, Berlin (2009)
Şahin, M.: Liftings of a monomial curve. Bull. Austral. Math. Soc. 98(2), 1–9 (2018)
Shen, B.: The parametric Frobenius problem and parametric exclusion. arXiv:1510.01349 (2015)
Sturmfels, B.: Gröbner Bases and Convex Polytopes, University Lecture Series, vol. 8, American Mathematical Soc. (1996)
Sullivant, S.: Algebraic Statistics, Graduate Studies in Mathematics, vol. 194, American Mathematical Soc. (2018)
Woods, K.: The unreasonable ubiquitousness of quasi-polynomials. Electron. J. Combin. 21(1), 1–44 (2014)
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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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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DOI: https://doi.org/10.1007/s00233-021-10164-3