Weighted sums of squares in local rings and their completions. I
type: Item
instance of: scholarly article
title: Weighted sums of squares in local rings and their completions. I
zbMATH Open document ID: 1198.14056
author: Claus Scheiderer
published in: Mathematische Zeitschrift
publication date: 2010-09-10T00:00:00Z
review text: Let \(\mathbb{R}[{\mathtt x}]=\mathbb{R}[x_1,\ldots, x_n],\) and let \(g_1,\ldots, g_k\in \mathbb{R}[{\mathtt x}]\) define the basic semialgebraic set \(K=\{x\in \mathbb{R}: g_i(x)\geq 0, i=1,\ldots,r\}.\) A natural and popular question is whether a polynomial \(f\in \mathbb{R}[{\mathtt x}]\) which is nonnegative on \(K\) lies in the preorder generated by \(g_1,\ldots, g_n,\) that is whether every such \(f\) has a representation \quad \((*): \; f=\sum_{i=1}^k a_i^2 h_i ,\) \quad where the \(h_i\) are products of some of the \(g_i,\) and the \(a_i\in \mathbb{R}[{\mathtt x}]\). Such representations are called nonnegativity certificates. \textit{C. Scheiderer} [Math. Z. 245, 725--760 (2003; Zbl 1056.14078)] has shown that, roughly speaking, an identity \((*)\) exists in \(\mathbb{R}[{\mathtt x}]\) iff it exists at sufficiently many localization \(\mathbb{R}[{\mathtt x}]_{\mathfrak{m}}\) by a maximal ideal \(\mathfrak{m}\). See [\textit{M. Marshall}, Positive Polynomials and Sums of Squares. chapter 9, Mathematical Surveys and Monographs 146. Providence, RI: American Mathematical Society (AMS). (2008; Zbl 1169.13001)] for a self-contained introduction to Scheiderer's local-global principles. In a more abstract setting, given a ring \(A\) and any subset \(M\subset A,\) define \(\mathcal{X}(M)=\{\alpha: \forall f\in M \; f(\alpha)\geq 0\},\) and the saturation of \(M\) by \(\text{Sat}(M)=\{f\in A: \forall \alpha\in \mathcal{X}(M) \; f(\alpha)\geq 0\}.\) Of course \(\text{Sat}(M)\supseteq M.\) So if \(T\) is a preorder generated by finitely many elements \(g_1,\dots,g_r\) in a local ring \(A\) the saturation Sat\((T)\) of \(T\) consists of all elements of \(A\) which are nonnegative where all \(g_i\) are nonnegative on the real spectrum \(\text{Sper}A\) of \(A.\) The existence of a nonnegativity certificate can now be seen to follow from answering for local rings \(A\) when is \(\text{Sat}(T)=T\)? \quad In this paper new light is shed onto the question by relating it to th…
reviewed by: Alexander Kovačec
Mathematics Subject Classification ID: 14P99
Mathematics Subject Classification ID: 13H05
Mathematics Subject Classification ID: 14P10
Mathematics Subject Classification ID: 32S05
Mathematics Subject Classification ID: 13B40
Mathematics Subject Classification ID: 13F40
Mathematics Subject Classification ID: 13J10
zbMATH DE Number: 5782746
zbMATH Keywords: local rings
zbMATH Keywords: completion
zbMATH Keywords: Artin approximation
zbMATH Keywords: preorderings
zbMATH Keywords: curve singularities
zbMATH Keywords: positive polynomials
zbMATH Keywords: sums of squares
zbMATH Keywords: real algebraic geometry
zbMATH Keywords: henselian ring
zbMATH Keywords: excellent ring
zbMATH Keywords: Krull topology
zbMATH Keywords: saturation
zbMATH Keywords: nonnegativity certificate
zbMATH Keywords: real spectrum
zbMATH Keywords: constructible topology
zbMATH Keywords: spectral topology
zbMATH Keywords: basic semialgebraic sets
zbMATH Keywords: local global principles
MaRDI profile type: Publication
full work available at URL: s00209-009-0551-6
OpenAlex ID: W2109980502
cites work: Q4889879
cites work: Algebraic approximation of structures over complete local rings
cites work: Le théorème des zéros pour les variétés analytiques réelles de dimension $2$
cites work: Q3701517
cites work: Q5619178
cites work: Positive semidefinite germs in real analytic surfaces
cites work: Positive semidefinite germs on the cone.
cites work: Sums of squares in real rings
cites work: Sums of squares of linear forms
cites work: Q4693114
cites work: Q3160258
cites work: Spaces of orderings and abstract real spectra
cites work: Q4510745
cites work: Q2716044
cites work: Sums of two squares in analytic rings
cites work: Q4843083
cites work: Sums of squares of regular functions on real algebraic varieties
cites work: On sums of squares in local rings
cites work: Sums of squares on real algebraic curves
cites work: Sums of squares on real algebraic surfaces
cites work: Non-existence of degree bounds for weighted sums of squares representations
cites work: Q3601991
DOI: 10.1007/S00209-009-0551-6
description: scientific article; zbMATH DE number 5782746