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.Comment: Final version, 36 pages. arXiv admin note: substantial text overlap
with arXiv:1310.130
