On Atomic Density of Numerical Semigroup Algebras

A numerical semigroup $S$ is a cofinite, additively-closed subset of the nonnegative integers that contains $0$. In this paper, 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 ${\mathbb{F}}_q[x]$ is zero for any finite field ${\mathbb{F}}_q$; we prove that the numerical semigroup algebra ${\mathbb{F}}_q[S]$ also has atomic density zero for any numerical semigroup~$S$. We also examine the particular algebra ${\mathbb{F}}_2[x^2,x^3]$ 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 M"{o}bius inversion, comparable to the formula for irreducible polynomials over a finite field ${\mathbb{F}}_q$.