Space of real arc germs and Poincaré series of a semi-algebraic set. (Espace des germes d’arcs réels et série de Poincaré d’un ensemble semi-algébrique.) (French) Zbl 0967.14037

The author investigates a real analogy of the theory developed by J. Denef and F. Loeser [Invent. Math. 135, 201-232 (1999; Zbl 0928.14004)]. First he defines arc spaces of germs of real arcs on semi-algebraic and real constructible sets. The geometry of these spaces yields new interesting invariants of semi-algebraic sets which are presented in the first section. Further on the author introduces a real version of the result of J. Pas on the \(p\)-adic cell decomposition [J. Reine Angew. Math. 399, 137-172 (1989; Zbl 0666.12014)] in order to obtain a theorem on quantifier elimination. The author applies the technics developed in sections 2 and 3 to the study of the Poincaré series for a semi-algebraic set and to prove that such a series belongs in fact to \(\mathbb Q[T]\).
Reviewer: Z.Hajto (Kraków)


14P10 Semialgebraic sets and related spaces
13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series
14G27 Other nonalgebraically closed ground fields in algebraic geometry
32B10 Germs of analytic sets, local parametrization
12L99 Connections between field theory and logic
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