Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 14-Day Trial for You or Your Team.

Learn More →

An Artin-Lang property for germs of C ∞ functions

An Artin-Lang property for germs of C ∞ functions Abstract. The rings of C y and analytic function germs on a real ane analytic germ V are studied, focusing on the basicness, separation and extension problems. Let A H V be a semianalytic set germ and assume there exist C y non-flat function germs ji (e.g. belonging to a quasi-analytic Denjoy-Carleman class) such that A ¼ fx A V j j1 ðxÞ > 0; . . . ; jk ðxÞ > 0g: then A is shown to be basic open in the analytic setting; an analogous statement concerning principal and separable set germs is proved similarly. These results are derived from an Artin-Lang-like property stating that a con~ structible set A f associated to A in the real spectrum of the ring of formal power series is ~ generically well-defined and that A ¼ j implies A f ¼ j. As a further consequence it is shown that given an analytic set germ X H V and a non-negative function germ g on X, g has an analytic non-negative extension to V if and only if it has a smooth non-negative extension. Introduction Consider the germ at the origin of the following semianalytic set: S ¼ fx http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal für die reine und angewandte Mathematik (Crelle's Journal) de Gruyter

An Artin-Lang property for germs of C ∞ functions

Loading next page...
 
/lp/de-gruyter/an-artin-lang-property-for-germs-of-c-functions-eedOwsR1cE

References (7)

Publisher
de Gruyter
Copyright
Copyright © 2002 by Walter de Gruyter GmbH & Co. KG
ISSN
0075-4102
eISSN
1435-5345
DOI
10.1515/crll.2002.056
Publisher site
See Article on Publisher Site

Abstract

Abstract. The rings of C y and analytic function germs on a real ane analytic germ V are studied, focusing on the basicness, separation and extension problems. Let A H V be a semianalytic set germ and assume there exist C y non-flat function germs ji (e.g. belonging to a quasi-analytic Denjoy-Carleman class) such that A ¼ fx A V j j1 ðxÞ > 0; . . . ; jk ðxÞ > 0g: then A is shown to be basic open in the analytic setting; an analogous statement concerning principal and separable set germs is proved similarly. These results are derived from an Artin-Lang-like property stating that a con~ structible set A f associated to A in the real spectrum of the ring of formal power series is ~ generically well-defined and that A ¼ j implies A f ¼ j. As a further consequence it is shown that given an analytic set germ X H V and a non-negative function germ g on X, g has an analytic non-negative extension to V if and only if it has a smooth non-negative extension. Introduction Consider the germ at the origin of the following semianalytic set: S ¼ fx

Journal

Journal für die reine und angewandte Mathematik (Crelle's Journal)de Gruyter

Published: Jun 12, 2002

There are no references for this article.