On the number of numerical semigroups containing two coprime integers \(p\) and \(q\).
type: Item
instance of: scholarly article
title: On the number of numerical semigroups containing two coprime integers \(p\) and \(q\).
zbMATH Open document ID: 1335.20058
published in: Semigroup Forum
publication date: 2015-10-12T00:00:00Z
review text: The manuscript under review relates numerical semigroups containing two coprime integers with rational convex polytopes and with lattice paths in the positive orthant of the plane. This enables the authors not only to determine the number of these objects, but also describe the shape of the function that counts them. Let \(p\) and \(q\) be coprime integers and denote by \(n(p,q)\) the number of numerical semigroups containing \(\{p,q\}\). Then fixing \(p\), there is a quasipolynomial of degree \(p-1\) taking the value \(n(p,q)\) for each \(q\in\mathbb N\) (the set of nonnegative integers), and the leading coefficient is constant. Moreover, bounds for the leading coefficient are given, and the value of \(n(p,q)\) is increasing in both variables. A numerical semigroup is a submonoid of \((\mathbb N,+)\) with finite complement in \(\mathbb N\). Having finite complement in \(\mathbb N\) is equivalent to having greatest common divisor of its elements equal to one. If we are given \(p\) and \(q\) coprime, then the submonoid they span is a numerical semigroup. Since the complement of this numerical semigroup is finite, there are only finitely many numerical semigroups containing \(p\) and \(q\); this is why \(n(p,q)\) is finite. The authors associate to each numerical semigroup containing \(p\) and \(q\) a point with nonnegative integer coordinates in a cone in \(\mathbb R^p\) intersected by the hyperplane \(x_p=q\). This is done via Apéry sets with respect to \(p\). Then they use Erhart's results to show that the number of integer points in that polytope is a quasipolynomial. In order to prove that the function is increasing in both variables and the properties of the leading coefficient, the authors relate to each point in the above polytope a lattice path in the plane from \((0,p)\) to \((q,0)\) with unit steps to the right and down. Examples, figures and explicit computations are given to illustrate the results.
Mathematics Subject Classification ID: 20M14
Mathematics Subject Classification ID: 52B05
Mathematics Subject Classification ID: 11P21
zbMATH DE Number: 6493128
zbMATH Keywords: numbers of numerical semigroups
zbMATH Keywords: Apéry sets
zbMATH Keywords: convex polytopes
zbMATH Keywords: quasipolynomials
zbMATH Keywords: lattice paths
author: Michael Hellus
author: R. Waldi
reviewed by: Pedro A. García Sánchez
describes a project that uses: polymake
MaRDI profile type: Publication
full work available at URL: s00233-015-9710-8
OpenAlex ID: W2041967853
cites work: polymake: a framework for analyzing convex polytopes
cites work: Q3754723
cites work: Geometrical illustration of numerical semigroups and of some of their invariants.
cites work: Q3903764
cites work: Lattice paths with given number of turns and semimodules over numerical semigroups
cites work: SYSTEMS OF INEQUALITIES AND NUMERICAL SEMIGROUPS
cites work: Q3225387
DOI: 10.1007/S00233-015-9710-8
description: scientific article; zbMATH DE number 6493128