Polynomials over ordered fields
type: Item
instance of: scholarly article
title: Polynomials over ordered fields
zbMATH Open document ID: 1111.12001
author: Constantin N. Beli
published in: Journal of Pure and Applied Algebra
publication date: 2007-02-26T00:00:00Z
review text: There is a theorem, published in 1928 by Pólya, about the representation of any real form \(F\) which is positive on the region \(\{(x_1,\ldots,x_n)\in \mathbb R^n : \text{\;all\;} x_j\geq0 \text{\;and\;} \sum x_j>0\}\) as a quotient \(G/H\), where \(G\) is a form all whose coefficients are positive and \(H=(x_1+\ldots+x_n)^p\), for some \(p\in\mathbb N\). Written as \(G/H\), the positivity of \(F\) is evident. The paper under review can be considered as a vast generalization of Pólya's theorem. In the first place, the real field is replaced by any ordered field \(K\). Let \(\overline K\) be the real closure of \(K\). Let \(X=(X_1,\ldots, X_n)\) be indeterminates. For any \(F\in K[X]\), let \(| F| \) denote the polynomial whose coefficients are the absolute values of the coefficients of \(F\). The author proves the following theorem (Theorem 3.1): given a form \(0\neq F\) in \(K[X]\), there exist forms \(G,H\) in \(K[X]\) all whose coefficients are positive so that \(FH=G\) if and only if there exists a rational \(\epsilon>0\) such that \(\epsilon | F| (x)< F(x)\), for all \(x\in \{(x_1,\ldots,x_n)\in {\overline K}^n : \text{\;all\;} x_j\geq0 \text{\;and\;} \sum x_j>0\}\). This result follows from the paper's main theorem: given a polynomial \(0\neq P\) in \(K[X]\), there exists a polynomial \(0\neq H\) in \(K[X]\) such that all the coefficients of \(FH\) are non-negative if and only if there exists a rational \(\epsilon>0\) such that \(\epsilon | F| (x)< F(x)\), for all \(x\in \{(x_1,\ldots,x_n)\in {\overline K}^n : \text{\;all\;} x_j\geq0 \}\). Moreover, if such an \(H\) exists, it can be chosen having all coefficients non-negative or having all coefficients non-positive. The techniques used in the proofs come from the theories of ordered fields and of convex sets and polytopes. As the author points out, Pólya's method does not apply to non-Archimedean fields. In addition to this paper, there are several papers about Pólya's theorem which might be interesting to t…
reviewed by: María Jesús de la Puente
Mathematics Subject Classification ID: 12D15
Mathematics Subject Classification ID: 11E10
Mathematics Subject Classification ID: 52B11
zbMATH DE Number: 5129955
zbMATH Keywords: polynomial
zbMATH Keywords: ordered field
zbMATH Keywords: Pólya's theorem
zbMATH Keywords: non-negative coefficients
zbMATH Keywords: positive coefficients
MaRDI profile type: Publication
full work available at URL: j.jpaa.2006.06.014
OpenAlex ID: W2071810978
cites work: Q5812325
cites work: Q4141914
cites work: Q5638112
DOI: 10.1016/J.JPAA.2006.06.014
description: scientific article; zbMATH DE number 5129955