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Quasi-polynomial growth of numerical and affine semigroups with constrained gaps.
- Source :
-
Semigroup Forum . Aug2023, Vol. 107 Issue 1, p60-78. 19p. - Publication Year :
- 2023
-
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. [ABSTRACT FROM AUTHOR]
- Subjects :
- *POLYHEDRA
*POLYTOPES
*GEOMETRY
*MULTIPLICITY (Mathematics)
*INTEGERS
*POLYNOMIALS
Subjects
Details
- Language :
- English
- ISSN :
- 00371912
- Volume :
- 107
- Issue :
- 1
- Database :
- Academic Search Index
- Journal :
- Semigroup Forum
- Publication Type :
- Academic Journal
- Accession number :
- 170040949
- Full Text :
- https://doi.org/10.1007/s00233-023-10366-x