Quantifier elimination in quasianalytic structures via non-standard analysis

TL;DR

This paper advances the theory of quantifier elimination in quasianalytic structures using non-standard analysis, providing new results and counterexamples that challenge classical theorems in real analytic geometry.

Contribution

It extends previous work by establishing a more general quantifier elimination framework and constructs a quasianalytic structure where classical theorems fail.

Findings

Quantifier elimination achieved in a broader quasianalytic setting.

Existence of a quasianalytic structure with non-semianalytic definable curves.

Counterexample to classical theorems on subanalytic and semianalytic sets.

Abstract

The paper is a continuation of our earlier article where we developed a theory of active and non-active infinitesimals and intended to establish quantifier elimination in quasianalytic structures. That article, however, did not attain full generality, which refers to one of its results, namely the theorem on an active infinitesimal, playing an essential role in our non-standard analysis. The general case was covered in our subsequent preprint, which constitutes a basis for the approach presented here. We also provide a quasianalytic exposition of the results concerning rectilinearization of terms and of definable functions from our earlier research. It will be used to demonstrate a quasianalytic structure corresponding to a Denjoy-Carleman class which, unlike the classical analytic structure, does not admit quantifier elimination in the language of restricted quasianalytic functions augmented merely by the reciprocal function. More precisely, we construct a plane definable curve, which indicates both that the classical theorem by J. Denef and L. van den Dries as well as \L{}ojasiewicz's theorem that every subanalytic curve is semianalytic are no longer true for quasianalytic structures. Besides rectilinearization of terms, our construction makes use of some theorems on power substitution for Denjoy-Carleman classes and on non-extendability of quasianalytic function germs. The last result relies on Grothendieck's factorization and open mapping theorems for (LF)-spaces. Note finally that this paper comprises our earlier preprints on the subject from May 2012.