Sur les cônes semi-analytiques. (On semi-analytic cones).

*(French)*Zbl 0606.32005The author gives here a proof of an observation made by him several years ago: ”A semi-analytic cone is semi-algebraic”. The fact has often been used during these years, since it provides an easy way of constructing subanalytic sets that are not semi-analytic (cones with transcendent base). However, the formal proof of the observation had not been written anywhere until the author in this paper and G. Raby (Poitiers) in his doctor’s thesis wrote, independently, two proofs similar in nature although different from the point of view of their presentation. The central observation (in both proofs) boils down to the fact that for the points of a dense subset of the given semi-analytic cone E, its germ \(E_ a\) is contained in an algebraic germ of the same dimension. That it is locally (for every point) contained in an analytic germ of the same dimension is known by the author’s classical theorem [in S. Łojasiewicz, Act. Congr. internat. Math. 1970, 2, 237-241 (1971; Zbl 0241.32005)]. The rest of the proof goes by induction on \(\dim E.\)

Reviewer: Z.Denkowska

##### MSC:

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