An effective version of Pólya's theorem on positive definite forms
type: Item
instance of: scholarly article
title: An effective version of Pólya's theorem on positive definite forms
zbMATH Open document ID: 0859.11029
DOI: 10.1016/0022-4049(95)00042-9
author: Jesús A. De Loera
author: Francisco Santos
published in: Journal of Pure and Applied Algebra
publication date: 1997-04-09T00:00:00Z
review text: Let \(S= \{x\in\mathbb R^n\mid x_1\geq 0, \ldots, x_n\geq 0,\, \sum x_i>0\}\). A classical result by Pólya asserts that any homogeneous polynomial \(F\in\mathbb R [X_1, \ldots, X_n]\) with \(F(x)>0\) for \(x\in S\) can be represented as a quotient \({G \over (X_1+ \cdots + X_n)^p}\) where all coefficients of \(G\in\mathbb R [X_1, \ldots, X_n]\) are positive. The least exponent \(p\) having this property is called the Pólya exponent. The authors prove a sharpened version of Pólya's theorem which supplies an upper bound for the Pólya exponent. If the coefficients of \(F\) are all rational then the upper bound involves the number of variables, the degree of \(F\) and the size of the coefficients of \(F\). The result is applied to find a representation of a positive definite polynomial \(F\in \mathbb R [X_1, \ldots, X_n]\) as a quotient of two sums of squares \(G,H\in \mathbb R [X_1, \ldots, X_n]\) (Hilbert's 17th problem).
reviewed by: Niels Schwartz
Mathematics Subject Classification ID: 11E10
Mathematics Subject Classification ID: 12J15
Mathematics Subject Classification ID: 13F20
zbMATH DE Number: 897489
zbMATH Keywords: Pólya's theorem on positive definite forms
zbMATH Keywords: ordered field
zbMATH Keywords: Hilbert's 17th problem
zbMATH Keywords: homogeneous polynomial
zbMATH Keywords: upper bound for the Pólya exponent
zbMATH Keywords: positive definite polynomials
zbMATH Keywords: sums of squares
MaRDI profile type: Publication
full work available at URL: 0022-4049(95)00042-9
OpenAlex ID: W2085112957
cites work: Q3770650
cites work: Q3135193
cites work: Q4039929
cites work: Uniform denominators in Hilbert's seventeenth problem
cites work: Q4079589
cites work: Effective Łojasiewicz inequalities in semialgebraic geometry
description: scientific article; zbMATH DE number 897489