On irreducible elements of numerical semigroup algebras

Abstract

Given a numerical semigroup S and a field F, the numerical semigroup algebra F[S] is a subalgebra of F[x] generated by monomials xs for s∈S. For a finite field Fq, we study distributions of irreducible elements of Fq[S]. We calculate the maximum number k of irreducible factors when an irreducible element of Fq[S] is factored in Fq[x]. We also show that the density of irreducible elements of degree n in Fq[S] has an asymptotic growth of (log⁡n)k−1/n. © 2026 Elsevier Inc.