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Geometric and differential properties of subanalytic sets. (English) Zbl 0912.32006
This exquisite paper is a continuation of the steady development of the theory of subanalytic sets treated in a very geometrical way by the authors [Publ. Math. Inst. Hautes Étud. Sci. 67, 5-42 (1988; Zbl 0674.32002)]. The main result of the present (very much self-contained) paper is Theorem 1.13 which states the equivalence of the following properties for a closed subanalytic subset of \(\mathbb{R}^n\):
(1) Composite function property
(2) Uniform Chevalley estimate
(3) Semicontinuity of the diagram of initial exponents
(4) Semicontinuity of the Hilbert-Samuel function
(5) Semicoherence.
It should be noticed that (1) is an old, unsolved, Thom-Glaeser problem and that semicoherence, particularly adapted to the problems in the real spaces, has been underestimated by other authors.
The work is a natural prolongation of the works of R. Thom, H. Hironaka, S. Łojasiewicz and A. M. Gabrielov.

32B20 Semi-analytic sets, subanalytic sets, and generalizations
32B15 Analytic subsets of affine space
Zbl 0674.32002
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