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.
Highlights
And a polynomial part of a partition function W (s, {d1, . . . , dm}), which gives the number of partitions of s ≥ 0 into m positive integers dj, and we put forward a conjecture about their relationship
In 2017, while studying the polynomial identities of arbitrary degree for syzygies degrees of the numerical semigroups d1, . . . , dm, we introduced a new kind of symmetric polynomials Pn(x1, . . . , xm) of degree n in m real variables xj: m m m
We study a factorization of Pn(xm) for n > m and by making use of this property, we find a representation of Pn(xm) through the power sums Ek =
Summary
Department of Civil Engineering, Technion – Israel Institute of Technology, Haifa 32000, Israel (Received: 24 November 2020. Received in revised form: 18 March 2021. Accepted: 19 March 2021. Published online: 22 March 2021.) c 2021 the author. This is an open access article under the CC BY (International 4.0) license (www.creativecommons.org/licenses/by/4.0/).
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