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Quasi-polynomial growth of numerical and affine semigroups with constrained gaps

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Abstract

A common tool in the theory of numerical semigroups is to interpret a desired class of semigroups as the lattice points in a rational polyhedron in order to leverage computational and enumerative techniques from polyhedral geometry. Most arguments of this type make use of a parametrization of numerical semigroups with fixed multiplicity m in terms of their m-Apéry sets, giving a representation called Kunz coordinates which obey a collection of inequalities defining the Kunz polyhedron. In this work, we introduce a new class of polyhedra describing numerical semigroups in terms of a truncated addition table of their positive sporadic elements. Applying a classical theorem of Ehrhart to slices of these polyhedra, we prove that the number of numerical semigroups with n sporadic elements and Frobenius number f is polynomial up to periodicity, or quasi-polynomial, as a function of f for fixed n. We also generalize this approach to higher dimensions to demonstrate quasi-polynomial growth of the number of affine semigroups with a fixed number of elements, and all gaps, contained in an integer dilation of a fixed polytope.

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Notes

  1. The semigroups described by the addition table \(\tau _0\) are the so-called elementary numerical semigroups (with n positive sporadic elements), defined as the semigroups with Frobenius number at most twice the multiplicity.

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Acknowledgements

We thank Nathan Kaplan for providing detailed feedback on our first draft, including suggesting additional connections to the semigroup literature and motivating the probabilistic interpretation of Corollary 2 given by Corollary 3. We also thank the anonymous referee for their comments and suggestions, which helped to improve the presentation of the paper.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of South Alabama, Mobile, AL, USA

    Michael DiPasquale

  2. Department of Mathematics, Colorado State University, Fort Collins, CO, USA

    Bryan R. Gillespie & Chris Peterson

Authors
  1. Michael DiPasquale
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  2. Bryan R. Gillespie
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  3. Chris Peterson
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Correspondence to Bryan R. Gillespie.

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Communicated by Nathan Kaplan.

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DiPasquale, M., Gillespie, B.R. & Peterson, C. Quasi-polynomial growth of numerical and affine semigroups with constrained gaps. Semigroup Forum 107, 60–78 (2023). https://doi.org/10.1007/s00233-023-10366-x

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