Prime divisors on real varieties and valuation theory
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.
scientific article; zbMATH DE number 3957251
scientific article