Prime divisors on real varieties and valuation theory
type: Item
instance of: scholarly article
title: Prime divisors on real varieties and valuation theory
zbMATH Open document ID: 0595.12015
DOI: 10.1016/0021-8693(86)90009-8
published in: Journal of Algebra
publication date: 1986-01-01T00:00:00Z
review text: The author studies three concepts over real function fields F/\({\mathbb{R}}\). In Section 1 he deals with the real holomorphy ring \(H_ F\) which is the intersection of all valuation rings of F having a formally real residue field. He proves that a set D of such valuation rings yields \(H_ F\) as intersection if and only if for every regular projective model W of F/\({\mathbb{R}}\) and every \({\mathbb{R}}\)-rational point \(P\in W\) some \(v\in D\) is centered in P. In particular he proves that the set of real prime divisors of F/\({\mathbb{R}}\) yields \(H_ F\) as intersection. In Section 2 the author gives a geometric description of signatures of higher level and, as an application, characterizes for even natural numbers n, which functions \(f\in F^{\times}\) are sums of n-th powers modulo \(H_ F^{\times}:\) A necessary and sufficient condition for f to belong to \(H_ F^{\times}\cdot \Sigma F^ n\) is that, with respect to all real prime divisors of F/\({\mathbb{R}}\), the value of f is n-divisible. In Section 3 the author proves that in the Bröcker-Prestel Local- Global-Principle for weak isotropy of quadratic forms over F/\({\mathbb{R}}\) [cf. the reviewer, Lectures on formally real fields (Lect. Notes Math. 1093) (1984; Zbl 0548.12010), Theorem 8.12] one may locally restrict the Henselizations to real prime divisors of F/\({\mathbb{R}}\). He also proves that in case of tr deg F/\({\mathbb{R}}>1\) none of the real prime divisors can be omitted.
reviewed by: Alexander Prestel
Mathematics Subject Classification ID: 12J10
Mathematics Subject Classification ID: 12D15
Mathematics Subject Classification ID: 11E10
Mathematics Subject Classification ID: 14Pxx
Mathematics Subject Classification ID: 13A18
zbMATH DE Number: 3957251
zbMATH Keywords: real function fields
zbMATH Keywords: real holomorphy ring
zbMATH Keywords: valuation rings
zbMATH Keywords: formally real residue field
zbMATH Keywords: regular projective model
zbMATH Keywords: rational point
zbMATH Keywords: real prime divisors
zbMATH Keywords: signatures of higher level
zbMATH Keywords: sums of n-th powers
zbMATH Keywords: Local-Global- Principle
zbMATH Keywords: weak isotropy
zbMATH Keywords: quadratic forms
zbMATH Keywords: Henselizations
author: Heinz-Werner Schülting
MaRDI profile type: Publication
full work available at URL: 0021-8693(86)90009-8
OpenAlex ID: W1985529310
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description: scientific article; zbMATH DE number 3957251