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- W3010596670 abstract "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 $mathbb{F}_q$." @default.
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- W3010596670 date "2020-03-03" @default.
- W3010596670 modified "2023-09-23" @default.
- W3010596670 title "On Atomic Density of Numerical Semigroup Algebras" @default.
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- W3010596670 doi "https://doi.org/10.48550/arxiv.2003.01710" @default.
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