Submission date: 26 December 2007. Revised versions: 10 April 2008 (with a new title), 19 May 2008.

Abstract:

This paper investigates the geometry of the expansion of the real field 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 Parusinski--Lion--Rolin). To this end, we study non-standard models of its universal diagram in the language augmented by the names of rational powers. Our approach makes no appeal to the Weierstrass preparation theorem, upon which 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 previous article, combines modifications by blowing up with a suitable partitioning.

Mathematics Subject Classification: Primary 32S45, 14P15, 32B20; Secondary 03C10, 26E10, 03C64.

Keywords and phrases: quasianalytic functions, special cubes, special modifications, analytically independent infinitesimals, active and non-active infinitesimals, valuation property, quantifier elimination, preparation theorem.

Full text: pdf, dvi, ps. (Version of 26 December 2007, title: Quantifier elimination, valuation property & preparation theorem in subanalytic geometry via transformation to normal crossings, pdf, dvi, ps. Version of 10 April 2008: pdf, dvi, ps.)