Quantifier elimination, valuation property and preparation theorem in quasianalytic geometry via transformation to normal crossings
Volume 96 / 2009
Abstract
This paper investigates the geometry of the expansion ${\cal R}_{Q}$ of the real field $\mathbb R$ by restricted quasianalytic functions. The main purpose is to establish quantifier elimination, description of definable functions by terms, the valuation property and preparation theorem (in the sense of Parusiński–Lion–Rolin). To this end, we study non-standard models $\cal R$ of the universal diagram $T$ of ${\cal R}_{Q}$ in the language $\cal L$ augmented by the names of rational powers. Our approach makes no appeal to the Weierstrass preparation theorem, upon which the majority of fundamental results in analytic geometry rely, but which is unavailable in the general quasianalytic geometry. The basic tools applied here are transformation to normal crossings and decomposition into special cubes. The latter method, developed in our earlier article [Ann. Polon. Math. 96 (2009), 65–74], combines modifications by blowing up with a suitable partitioning. Via an analysis of $\cal L$-terms and infinitesimals, we prove the valuation property for functions given by $\cal L$-terms, and next the exchange property for substructures of a given model $\cal R$. Our proofs are based on the concepts of analytically independent as well as active and non-active infinitesimals, introduced in this article. Further, quantifier elimination for $T$ is established through model-theoretic compactness. The universal theory $T$ is thus complete and o-minimal, and ${\cal R}_{Q}$ is its prime model. Under the circumstances, every definable function is piecewise given by $\cal L$-terms, and therefore the previous results concerning $\cal L$-terms generalize immediately to definable functions. In this fashion, we obtain the valuation property and preparation theorem for quasi-subanalytic functions. Finally, a quasi-subanalytic version of Puiseux's theorem with parameter is demonstrated.
