zbMATH — the first resource for mathematics

\(w_ f\)-stratification of subanalytic functions and the Łojasiewicz inequality. (English. Abridged French version) Zbl 0799.32007
The authors show that for every subanalytic function \(f\) there is a stratification compatible with it and satisfying the Thom \(w_ f\) condition.
The methods used are the standard methods of the Lojasiewicz school, to which one of the authors (K. Kurdyka) belonged long enough to be aware of the fact that quoting S. Łojasiewicz [Semin. Geom., Univ. Studi Bologna 1986, 83-97 (1988; Zbl 0673.57006)] instead of Łojasiewicz’s “Ensembles semianalytiques”, preprint IHES 1967 is misleading, as well as quoting J.-L. Verdier [Invent. Math. 36, 295-312 (1976; Zbl 0333.32010)], but omitting the results of the reviewer, K. Wachta or M. Coste, M. Coste-Roy or T. C. Kuo about \((w)\) stratification, very close in spirit to the reviewed paper.
The good part of the paper are the ramifications, it is almost a survey about not enough known brillant works of Bekka, Mihalache etc.
The reviewer likes the links with the Łojasiewicz inequality as well as the remarks about the deep results obtained by A. Parusin’ski (the second author) concerning the Lipschitz properties of semianalytic sets [A. Parusinski, Ann. Inst. Fourier 33, No. 4, 189-213 (1988; Zbl 0661.32015)].

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