论文标题
数值半群代数的原子密度
On Atomic Density of Numerical Semigroup Algebras
论文作者
论文摘要
数值半群$ s $是包含$ 0 $的非负整数的cofinite，添加封闭的子集。在本文中，我们启动了原子密度的研究，这是对给定环或半群中不可约元素比例的渐近度量，用于半群代数。众所周知，对于任何有限字段$ \ mathbb {f} _q $，多项式环$ \ mathbb {f} _q [x] $的原子密度为零；我们证明，对于任何数值semigroup〜 $ s $，数值半群代数$ \ mathbb {f} _q [s] $也具有零。我们还可以更详细地检查特定代数$ \ Mathbb {f} _2 [x^2，x^3] $，从而提供了原子密度的收敛速度，以及使用Möbius倒入Möbius逆转的不可及的多种元素的计数公式，可与irreducible for Irreducile fightiene fordiene fornite fornite fornite fornite fornite fornite fornite fornite fornite fornite fornite fornite fornite fornite pitinials alinite pittions $ \ mathbb {f} _q $。
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 Möbius inversion, comparable to the formula for irreducible polynomials over a finite field $\mathbb{F}_q$.
