On a global analytic Positivstellensatz
type: Item
instance of: scholarly article
title: On a global analytic Positivstellensatz
zbMATH Open document ID: 1216.14053
author: José F. Fernando
author: Francesca Acquistapace
author: Fabrizio Broglia
published in: Arkiv för Matematik
publication date: 2009-09-14T00:00:00Z
review text: The authors study Positivstellensätze for global analytic functions. They consider an open subset \(\Omega\subseteq\mathbb{R}^n\) and an analytic set \(X\subseteq\Omega\), as well as real analytic functions \(g_1, \dots,g_r\), \(m:X\to\mathbb{R}\). Let \(Y\) be the zero set of \(M\) and let \(S= \{x\in X\mid\forall i= 1,\dots, r: g_i(x)\geq 0\}\). The authors ask whether the following two conditions are equivalent: \(\bullet\) \(S\subseteq Y\). \(\bullet\) There are sums of squares \(s_\nu\) of analytic functions with zero set contained in \(Y\) such that \(\sum_{\nu\in\{0, 1\}^r} s_\nu\cdot g^\nu= 0\). (The sums of squares may be finite or infinite.) The answer is affirmative if \(X\) is a curve (in this case the \(s_\nu\) are sums of 2 squares), or if \(X\) is a normal surface (in this case the \(s_\nu\) are sums of 5 squares), or if \(X\) is a real coherent set whose connected components are all compact.
Mathematics Subject Classification ID: 14P15
Mathematics Subject Classification ID: 14P10
Mathematics Subject Classification ID: 13J30
zbMATH DE Number: 5603927
zbMATH Keywords: real analytic set
zbMATH Keywords: analytic function
zbMATH Keywords: meromorphic function
zbMATH Keywords: Positivstellensatz
zbMATH Keywords: sums of quares
zbMATH Keywords: positive semidefinite function
reviewed by: Niels Schwartz
MaRDI profile type: Publication
full work available at URL: s11512-008-0082-0
OpenAlex ID: W2132597741
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DOI: 10.1007/S11512-008-0082-0
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description: scientific article; zbMATH DE number 5603927