Quantifier elimination, valuation property and preparation theorem in quasianalytic geometry via transformation to normal crossings

Krzysztof Jan Nowak¹

¹ Institute of Mathematics Jagiellonian University /Lojasiewicza 6 30-348 Kraków, Poland

Annales Polonici Mathematici, Tome 96 (2009) no. 3, pp. 247-282.

Résumé

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.

Détail

DOI : 10.4064/ap96-3-5

Keywords: paper investigates geometry expansion cal real field mathbb restricted quasianalytic functions main purpose establish quantifier elimination description definable functions terms valuation property preparation theorem sense parusi ski lion rolin end study non standard models cal universal diagram cal language cal augmented names rational powers approach makes appeal weierstrass preparation theorem which majority fundamental results analytic geometry rely which unavailable general quasianalytic geometry basic tools applied here transformation normal crossings decomposition special cubes latter method developed earlier article ann polon math combines modifications blowing suitable partitioning via analysis cal l terms infinitesimals prove valuation property functions given cal l terms exchange property substructures given model cal proofs based concepts analytically independent active non active infinitesimals introduced article further quantifier elimination established through model theoretic compactness universal theory complete o minimal cal its prime model under circumstances every definable function piecewise given cal l terms therefore previous results concerning cal l terms generalize immediately definable functions fashion obtain valuation property preparation theorem quasi subanalytic functions finally quasi subanalytic version puiseuxs theorem parameter demonstrated