Abstract
A numerical semigroup is a cofinite, additively closed subset of the nonnegative integers that contains . 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 is zero for any finite field ; we prove that the numerical semigroup algebra also has atomic density zero for any numerical semigroup . We also examine the particular algebra 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 .
Citation
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
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