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Lemmas A and B for sub-Pfaffian sets. (English) Zbl 0956.32008
Lemma A and lemma B are two popular code names for the fundamental lemmas proved by Z. Denkowska, S. Łojasiewicz, and J. Stasica in [Bull. Acad. Pol. Sci., Sér. Sci. Math. 27, 529-536 (1979; Zbl 0435.32006) and ibid., 537-539 (1979; Zbl 0457.32003)] and largely used to prove properties of subanalytic sets by Łojasiewicz’s semi-analytic methods.
The lemmas are about simplifying as much as possible the semi-analytic object we project to get our subanalytic set.
Pfaffian sets, introduced by R. Moussu and C. Roche [Invent. Math. 105, No. 2, 431-441 (1991; Zbl 0769.58050) and Ann. Inst. Fourier, 42, No. 1-2, 393-420 (1992; Zbl 0759.32005)] following an idea of Hovanski, are, roughly speaking, “good” solutions of analytic differential equations and are more general than semi-analytic sets, while keeping many of their properties (C. Roche, J. P. Lion). Sub-Pfaffian sets are their proper analytic images.
The present work is a nice survey on the subject. Analogues of Lemma A and B are given for sub-Pfaffian sets. Their proofs are mainly due to the author.
Good reading and a useful survey.
32B20 Semi-analytic sets, subanalytic sets, and generalizations
53C12 Foliations (differential geometric aspects)
34C30 Manifolds of solutions of ODE (MSC2000)