Winter 2022 On atomic density of numerical semigroup algebras
Austin A. Antoniou, Ranthony A. C. Edmonds, Bethany Kubik, Christopher O’Neill, Shannon Talbott
J. Commut. Algebra 14(4): 455-470 (Winter 2022). DOI: 10.1216/jca.2022.14.455

Abstract

A numerical semigroup S is a cofinite, additively closed subset of the nonnegative integers that contains 0. We initiate the study of atomic density, an asymptotic measure of the proportion of irreducible elements in a given ring or semigroup, for semigroup algebras. It is known that the atomic density of the polynomial ring 𝔽q[x] is zero for any finite field 𝔽q; we prove that the numerical semigroup algebra 𝔽q[S] also has atomic density zero for any numerical semigroup S. We also examine the particular algebra 𝔽2[x2,x3] in more detail, providing a bound on the rate of convergence of the atomic density as well as a counting formula for irreducible polynomials using Möbius inversion, comparable to the formula for irreducible polynomials over a finite field 𝔽q.

Citation

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Austin A. Antoniou. Ranthony A. C. Edmonds. Bethany Kubik. Christopher O’Neill. Shannon Talbott. "On atomic density of numerical semigroup algebras." J. Commut. Algebra 14 (4) 455 - 470, Winter 2022. https://doi.org/10.1216/jca.2022.14.455

Information

Received: 21 August 2020; Revised: 3 March 2021; Accepted: 5 March 2021; Published: Winter 2022
First available in Project Euclid: 15 November 2022

MathSciNet: MR4509402
zbMATH: 1516.13001
Digital Object Identifier: 10.1216/jca.2022.14.455

Subjects:
Primary: 12E05 , 13A05 , 20M14

Keywords: atomic density , finite field , numerical semigroup

Rights: Copyright © 2022 Rocky Mountain Mathematics Consortium

Vol.14 • No. 4 • Winter 2022
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