Representation of positive semidefinite elements as sum of squares in 2-dimensional local rings
type: Item
instance of: scholarly article
title: Representation of positive semidefinite elements as sum of squares in 2-dimensional local rings
zbMATH Open document ID: 1487.14125
author: José F. Fernando
published in: Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A: Matemáticas. RACSAM
publication date: 2022-01-24T00:00:00Z
review text: For an arbitrary commutative ring \(A\), the set \(\mathcal{P}(A)\) of positive semidefinite elements in \(A\) can be defined by using the real spectrum of \(A\). Let \(\Sigma A^2\) denote the set of finite sums of squares in \(A\). Comparing the two sets \(\mathcal{P}(A)\) and \(\Sigma A^2\) is a classical problem both in the theory of quadratic forms and in real geometry. When \(A\) is a henselian excellent local ring, it is known that the equality \(\mathcal{P}(A)=\Sigma A^2\) implies \(\dim A\le 2\). A natural question is to know which rings \(A\) of dimension \(\le 2\) have the property \(\mathcal{P}(A)=\Sigma A^2\). The 1-dimensional case was completely solved by \textit{C. Scheiderer} [J. Reine Angew. Math. 540, 205--227 (2001; Zbl 0991.13014)]. The paper under review considers the case where \(A\) is further assumed to have dimension 2 and embedding dimension 3. When the residue field \(\kappa\) of \(A\) is real closed, the author has determined in previous works (see [\textit{J. F. Fernando}, Math. Ann. 322, No. 1, 49--67 (2002; Zbl 1006.32008); Math. Z. 244, No. 4, 725--752 (2003; Zbl 1052.14069)]) a necessary and sufficient condition for the equality \(\mathcal{P}(A)=\Sigma A^2\) in terms of the analytic equivalence class (or equivalently, the structure of the completion \(\hat{A}\)) of \(A\). In the present paper there are two main theorems. In the first one (Theorem 1.5), the residue field \(\kappa\) is only assumed to be formally real. The result is that if \(\mathcal{P}(A)=\Sigma A^2\), then \(\hat{A}\) must be one of the candidates listed in the theorem. In the second main result (Theorem 1.8), one assumes that \(\kappa\) is formally real and that \(\tau(\kappa)<+\infty\). (The latter condition means that the level \(s(F)\) of every finite non-real extension \(F/\kappa\) is uniformly bounded from the above.) In this case, it is proved that when \(\hat{A}\) belongs to a given list the equality \(\mathcal{P}(A)=\Sigma A^2\) holds. As a corollary, when …
reviewed by: Yong Hu
Mathematics Subject Classification ID: 14P10
Mathematics Subject Classification ID: 11E25
Mathematics Subject Classification ID: 11E08
Mathematics Subject Classification ID: 12D15
Mathematics Subject Classification ID: 13F25
Mathematics Subject Classification ID: 13F40
Mathematics Subject Classification ID: 32B05
zbMATH DE Number: 7462213
zbMATH Keywords: positive semidefinite elements
zbMATH Keywords: sums of squares
zbMATH Keywords: real spectrum
zbMATH Keywords: singularities
zbMATH Keywords: excellent henselian ring
zbMATH Keywords: dimension 2
zbMATH Keywords: completion
MaRDI profile type: Publication
full work available at URL: s13398-021-01202-4
OpenAlex ID: W4206674210
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DOI: 10.1007/S13398-021-01202-4
description: scientific article; zbMATH DE number 7462213