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Tarski’s problem and Pfaffian functions. (English) Zbl 0616.03018
Logic colloq. ’84, Proc. Colloq., Manchester/U.K. 1984, Stud. Logic Found. Math. 120, 59-90 (1986).
[For the entire collection see Zbl 0587.00004.]
This is an update of the author’s ”Remarks on Tarski’s problem concerning \(({\mathbb{R}},+,\cdot,\exp)''\) [Logic colloquium ’82, Proc. Colloq., Florence 1982, Stud. Logic Found. Math. 112, 97-121 (1984; Zbl 0585.03006)]. The major new tool is the use of Pfaffian functions and cells, inspired by A. G. Khovanski’s ”On a class of systems of transcendental equations” [Dokl. Akad. Nauk SSSR 255, 804-807 (1980; Zbl 0569.32004)]. A minor new tool is (a hybrid application of) nonstandard analysis. The new ”ultimate goal” is the Decomposition Conjecture: The zero set of a Pfaffian function is a finite disjoint union of Pfaffian cells. The author describes how far one got and how he plans to proceed further.
Reviewer: G.Fuhrken

03C10 Quantifier elimination, model completeness and related topics
26B99 Functions of several variables