Symmetric polynomials associated with numerical semigroups

L. Fel
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引用次数: 1

Abstract

We study a new kind of symmetric polynomials P_n(x_1,...,x_m) of degree n in m real variables, which have arisen in the theory of numerical semigroups. We establish their basic properties and find their representation through the power sums E_k=\sum_{j=1}^m x_j^k. We observe a visual similarity between normalized polynomials P_n(x_1,...,x_m)/\chi_m, where \chi_m=\prod_{j=1}^m x_j, and a polynomial part of a partition function W(s,{d_1,...,d_m}), which gives a number of partitions of s\ge 0 into m positive integers d_j, and put forward a conjecture about their relationship.
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与数值半群相关的对称多项式
研究了在数值半群理论中出现的m个实数变量上的一类新的n次对称多项式P_n(x_1,…,x_m)。我们建立了它们的基本性质,并通过幂和E_k= \sum _j=1^m x_j{^k找到了它们的表示。我们观察到归一化多项式P_n(x_1,…,x_m)/ }\chi _m(其中\chi _m= \prod _j=1^m x_j{)与配分函数W(s,d_1,…,d_m)的多项式部分在视觉上的相似性,W(s,}d_1,…,d_m{)给出了s }\ge 0的若干划分为m个正整数d_j,并提出了它们之间的关系猜想。
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