Semi-analyticité et sous-analyticité. (Semi-analyticity and sub- analyticity) (Q1080991)

Given a subanalytic set D, its subsets \(D_{ns}\) and \(D_{nN}\) are considered, defined as follows: \[ D_{ns}=\{x\in D: \text{ the germ \(D_ x\) is not semi-analytic}\}, \] \[ D_{nN}=\{x\in D: \text{ the germ \(D_ x\) is not Nash subanalytic}\}. \] Nash subanalytic sets, introduced and studied by Bierstone, Milman and Schwarz as a very useful tool for some important problems of differential analysis (composition, division) may be defined as images of proper, analytic, regular in the sense of Gabrielov mappings (one of the equivalent definitions). In the paper the authors study the relations between \(D_{ns}\) and \(D_{nN}\) (for example: \(D_{ns}=D_{nN}\cup (\sin g D)_{nN})\) as well as the structure of these sets in connection with other questions. (The subanalyticity of \(D_{ns}\) and \(D_{nN}\) was proved later by Pawłucki, Kraków).