Polynomial representations via spectral decompositions
type: Item
instance of: scholarly article
title: Polynomial representations via spectral decompositions
zbMATH Open document ID: 1049.26005
DOI: 10.1023/A:1026213712849
author: Olivier Demanze
published in: Positivity
publication date: 2004-01-06T00:00:00Z
review text: The present note is an improvement and a continuation of [\textit{O. Demanze}, J. Math. Anal. Appl. 247, No.~2, 570--587 (2000; Zbl 0961.26007)]. Specifically, the author studies a polynomial \(P(x)\) depending on \(n\) real variables, which is strictly positive (in a strong sense, involving points at infinity in the projective completion) on a basic semialgebraic set \[ \{x\in\mathbb{R}^n;\, P_1(x)\geq 0,\dots, P_k(x)\geq 0\} \] defined by polynomials \(P_j\). The main result asserts that such a polynomial \(P\) can be represented as a weighted sum of squares of rational functions in the variables \(x\) and \(\sqrt{1+ x^2_j}\), \(1\leq j\leq n\), with a common denominator of the form \[ \prod^n_{j=1} \sqrt{1+ x^{2^{m_j}}},\quad m_j\geq 0. \] This is a refinement of known Positivstellensätze in real algebraic geometry. The proof is an adaptation of the functional theoretic technique contained in [\textit{M. Putinar} and \textit{F.-H. Vasilescu}, ``Solving moment problems by dimensional extension'', Ann. Math. (2) 149, No.~3, 1087--1107 (1999; Zbl 0939.44003)].
reviewed by: Mihai Putinar
Mathematics Subject Classification ID: 26C05
Mathematics Subject Classification ID: 14P10
Mathematics Subject Classification ID: 12D15
Mathematics Subject Classification ID: 44A60
zbMATH DE Number: 2021989
zbMATH Keywords: moment problem
zbMATH Keywords: positive polynomial
zbMATH Keywords: semi-algebraic set
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 2021989