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Complements of subanalytic sets and existential formulas for analytic functions. (English) Zbl 0851.32009
The author is the same person who started, even before Hironaka, the theory of subanalytic sets in 1968 [the author, Funct. Anal. Appl. 2, 282-291 (1968); translation from Funkts. Anal. Prilozh. 2, No. 4, 18-30 (1968; Zbl 0179.08503)]. In the cited paper he proved, among other things, the essential result: the complement of a subanalytic set in \(\mathbb{R}^n\) is subanalytic. Now he proves the same result for the subanalytic sets described by a subalgebra of analytic functions, closed under differentiation. The translation into the logic is given, too. Needless to say, such a result is not easy to obtain and the author uses the newest developments (like A. G. Khovanskij [‘Fewnomials’ (1991; Zbl 0728.12002)], for example). The paper is short and well written.

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