Quantifier elimination in quasianalytic structures via non-standard analysis

Krzysztof Jan Nowak¹

¹ Institute of Mathematics Faculty of Mathematics and Computer Science Jagiellonian University Łojasiewicza 6 30-348 Kraków, Poland

Annales Polonici Mathematici, Tome 114 (2015) no. 3, pp. 235-267.

Résumé

The paper is a continuation of an earlier one 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 quasianalytic 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 $1/x$. More precisely, we construct a definable plane curve, which indicates that both the classical theorem by J. Denef and L. van den Dries as well as Łojasiewicz'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.

Détail

DOI : 10.4064/ap114-3-4

Keywords: paper continuation earlier where developed theory active non active infinitesimals intended establish quantifier elimination quasianalytic structures article however did attain full generality which refers its results namely theorem active infinitesimal playing essential role non standard analysis general covered subsequent preprint which constitutes basis approach presented here provide quasianalytic exposition results concerning rectilinearization terms definable functions earlier research demonstrate quasianalytic structure corresponding quasianalytic denjoy carleman class which unlike classical analytic structure does admit quantifier elimination language restricted quasianalytic functions augmented merely reciprocal function precisely construct definable plane curve which indicates classical theorem nbsp denef nbsp van den dries ojasiewiczs theorem every subanalytic curve semianalytic longer quasianalytic structures besides rectilinearization terms construction makes theorems power substitution denjoy carleman classes non extendability quasianalytic function germs result relies grothendiecks factorization mapping theorems spaces