On the positivity of symmetric polynomial functions. I: General results
type: Item
instance of: scholarly article
title: On the positivity of symmetric polynomial functions. I: General results
zbMATH Open document ID: 1031.05130
DOI: 10.1016/S0022-247X(03)00301-9
author: Vlad Timofte
published in: Journal of Mathematical Analysis and Applications
publication date: 2003-09-04T00:00:00Z
review text: Let \(f(X_1,\dots,X_n)\) be a symmetric polynomial of degree \(d\geq 2\). The author studies the problem when \(f(X_1,\dots,X_n)\geq 0\) for all \((X_1,\dots,X_n)\in {\mathbb R}_+^n\). The main result is that this inequality holds if and only if it holds for all elements with at most \(\lfloor d/2\rfloor\) (the integer part) distinct non-zero components (which may have multiplicities). In order to establish this result the author uses that \(f\) can be expressed in terms of the symmetric power sums and solves a Cauchy problem for ordinary differential equations involving the symmetric power sums. This implies the existence of a special kind of paths in the minimizer of a restriction of the considered polynomial. The author obtains also analogues of the main results for inequalities which hold for polynomials of even degree and for all \((X_1,\dots,X_n)\in {\mathbb R}^n\).
Mathematics Subject Classification ID: 05E05
zbMATH DE Number: 1974966
zbMATH Keywords: symmetric function
zbMATH Keywords: symmetric power sums
zbMATH Keywords: minimizer
reviewed by: Vesselin Drensky
MaRDI profile type: Publication
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full work available at URL: s0022-247x(03)00301-9
OpenAlex ID: W2089478560
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description: scientific article; zbMATH DE number 1974966