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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 für die reine und angewandte Mathematik (Crelle's Journal) – de Gruyter
Published: Jun 12, 2002
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