Points de Nash des ensembles sous-analytiques. (Nash points of sub- analytic sets).

*(French)*Zbl 0694.32002
Mem. Am. Math. Soc. 425, 76 p. (1990).

A subanalytic set was defined by Gabrielov as being locally a projection of a relatively compact semianalytic set. The class of subanalytic sets is thus bigger than that of semianalytic sets. Some points of a subanalytic set may still be the points of semianalycity (in fact the “majority” of them is). The question arose (Hironaka, Łojasiewicz) whenever the set of semianalytic points of a subanalytic set is subanalytic. This was translated into an equivalent problem by Bierstone and Milman, who introduced the notion of Nash points. A point a of a subanalytic set Y of pure dimension k is said to be a Nash point if the smallest analytic germ in a containing the germ of Y is of dimension k. By the classical theorem of Łojasiewicz, all points of a semianalytic set are Nash points, while several examples show that this is not true for subanalytic sets. Bierstone and Milman showed that the set of Nash points of a subanalytic set is subanalytic if and only if the set of semianalytic points is. The author of the present paper proves: Theorem. The set of Nash points of a subanalytic set is subanalytic.

The proof is not easy. It is based mainly on the reasoning of Gabrielov from “Relations between analytic functions”, suitably adapted, and uses, naturally, the whole theory of subanalytic sets.

The proof is not easy. It is based mainly on the reasoning of Gabrielov from “Relations between analytic functions”, suitably adapted, and uses, naturally, the whole theory of subanalytic sets.

Reviewer: Z.Denkowska

##### MSC:

32B20 | Semi-analytic sets, subanalytic sets, and generalizations |