Solution of a problem about symmetric functions
type: Item
instance of: scholarly article
title: Solution of a problem about symmetric functions
zbMATH Open document ID: 1062.12005
author: Roberto Dvornicich
author: Umberto Zannier
published in: Rocky Mountain Journal of Mathematics
publication date: 2004-10-01T00:00:00Z
review text: Let \(X\), \(Y\) be independent indeterminates over \(\mathbb{Q}\), the field of rational numbers, and let \(m\) be a positive integer. Let \(N_m= N_m(X, Y)=X^m+ Y^m\) be the Newton symmetric power of order \(m\). The purpose of this paper is to prove the following result: If \(a> b> c\) are positive integers with \((a,b,c)= 1\), then the functions \(N_a\), \(N_b\), \(N_c\) generate the field \(S\) of all symmetric rational functions in \(X\), \(Y\) with rational coefficients, i.e., \(S= \mathbb{Q}(N_a, N_c)\). The authors then go on to prove a conjecture about the minimal degree \(d\) of a polynomial relation satisfied by \(N_a\), \(N_b\), \(N_c\), where by degree of a monomial \(N^i_a N^j_b N^k_c\), they mean \(ai+ bj+ ck\), namely \(d= abc/2\) if \(abc\) is even and \(d= (a- 1)bc/2\) otherwise. The authors supply some geometric applications.
Mathematics Subject Classification ID: 12F20
Mathematics Subject Classification ID: 05E05
zbMATH DE Number: 2104668
zbMATH Keywords: symmetric rational functions
zbMATH Keywords: Newton symmetric power
reviewed by: John N. Mordeson
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1216/rmjm/1181075462
OpenAlex ID: W2077236422
DOI: 10.1216/RMJM/1181075462
description: scientific article; zbMATH DE number 2104668