On orderings in real surfaces
type: Item
instance of: scholarly article
title: On orderings in real surfaces
zbMATH Open document ID: 0559.14015
DOI: 10.1016/0022-4049(85)90059-3
published in: Journal of Pure and Applied Algebra
publication date: 1985-01-01T00:00:00Z
review text: It is well-known that if C is an algebraic curve over the real closed field R and \(\alpha\) is a total ordering of the function field R(C) of C then there is a semi-algebraic embedding \(w: (0,1)\to C\) such that \(f\in R(C)\) is positive with respect to \(\alpha\) if and only if there is some \(t\in R\), \(0<t\) such that fw is defined and positive on (0,t). In the present paper it is shown that the total orderings of the function field of an algebraic surface over the field \({\mathbb{R}}\) of real numbers admits a similar geometric description. Let V be an irreducible algebraic surface over \({\mathbb{R}}\) embedded in some \({\mathbb{R}}^ n\). Using a discussion of the orderings of the meromorphic function germs of an irreducible analytic surface germ the following is proved: If \(\alpha\) is a total ordering of \({\mathbb{R}}(V)\) then there is an analytic map \(c: (0,1)\to V\) such that \(f\in {\mathbb{R}}(V)\) is positive with respect to \(\alpha\) if and only if fc is defined and positive on (0,t) for some \(0<t\in {\mathbb{R}}.\)
Mathematics Subject Classification ID: 14Pxx
Mathematics Subject Classification ID: 14J20
Mathematics Subject Classification ID: 12J15
zbMATH DE Number: 3891514
zbMATH Keywords: total orderings of the function field of an algebraic surface
zbMATH Keywords: meromorphic function germs
author: M. E. Alonso
author: Jose Manuel Gamboa
author: Jesús M. Ruiz
reviewed by: Niels Schwartz
MaRDI profile type: Publication
full work available at URL: 0022-4049(85)90059-3
OpenAlex ID: W2057549942
cites work: Q4057338
cites work: Q4037348
cites work: Q5509706
cites work: Q5590867
cites work: Q5582147
cites work: Q3340956
cites work: Central orderings in fields of real meromorphic function germs
description: scientific article; zbMATH DE number 3891514