On atomic density of numerical semigroup algebras

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.