Impossibility of extending Pólya's theorem to ``forms'' with arbitrary real exponents
type: Item
instance of: scholarly article
title: Impossibility of extending Pólya's theorem to ``forms'' with arbitrary real exponents
zbMATH Open document ID: 1155.13016
author: Charles N. Delzell
published in: Journal of Pure and Applied Algebra
publication date: 2008-10-21T00:00:00Z
review text: The starting point of the paper under review is the well-known and important Theorem of Polya. Let \(F\) be a homogeneous polynomial in \(n\) variables such that \(F>0\) on \(\mathbb{R}_{\geq 0}\setminus{\{0\}}\). Then the following holds. Strong form of Polya's Theorem: For all sufficiently large \(p\in\mathbb{N}\) the homogeneous polynomial \((X_1+\ldots+X_n)^p F(X_1,\ldots,X_n)\) has strictly positive coefficients (which means that the coefficients at the monomials of the appropriate homogeneous degree are strictly positive). Weak form of Polya's Theorem: \(F\) is the quotient of two homogeneous polynomials with every coefficient strictly positive. The author discusses the extension of the results above to homogeneous generalized polynomials with arbitrary real exponents. He shows the following: - The strong form of Polya's Theorem does not extend to generalized polynomials with non-integer exponents. - The weak form of Polya's Theorem extends to generalized polynomials with rational exponents. - The weak form of Polya's Theorem does not extend to generalized polynomials with arbitrary real exponents. The paper is carefully written and very detailed. A lot of information about the background and the literature is given.
reviewed by: Tobias Kaiser
Mathematics Subject Classification ID: 13J30
Mathematics Subject Classification ID: 12D99
Mathematics Subject Classification ID: 14P15
Mathematics Subject Classification ID: 26C05
zbMATH DE Number: 5354969
zbMATH Keywords: Polya's Theorem
zbMATH Keywords: generalized polynomials
MaRDI profile type: Publication
full work available at URL: j.jpaa.2008.04.006
OpenAlex ID: W1982433837
cites work: Polynomials over ordered fields
cites work: Q4858543
cites work: The elementary theory of restricted analytic fields with exponentiation
cites work: Q5557595
cites work: Geometric programming with signomials
cites work: Q3681045
cites work: A PTAS for the minimization of polynomials of fixed degree over the simplex
cites work: Théorème de préparation pour les fonctions logarithmico-exponentielles. (Preparation theorem for logarithmico-exponential functions)
cites work: Expansions of the real field with power functions
cites work: Q4329397
cites work: Generalized Polynomial Optimization
cites work: A new bound for Pólya's theorem with applications to polynomials positive on polyhedra.
cites work: Q2716044
cites work: Q5638112
cites work: Global optimization of nonconvex polynomial programming problems having rational exponents
cites work: Q4781203
cites work: A Simple Proof of Descartes's Rule of Signs
cites work: Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function
cites work: A MATHEMATICAL AID IN OPTIMIZING ENGINEERING DESIGNS
DOI: 10.1016/J.JPAA.2008.04.006
description: scientific article; zbMATH DE number 5354969