Quantifier elimination in quasianalytic structures via non-standard analysis

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

On division of quasianalytic function germs

In this paper, we establish the following criterion for divisibility in the local ring of those quasianalytic function germs at zero which are definable in a polynomially bounded structure. A sufficient (and necessary) condition for the divisibility of two such function germs is that of their Taylor series at zero in the formal power series ring

A closedness theorem and applications in geometry of rational points over Henselian valued fields

We develop geometry of algebraic subvarieties of $K^{n}$ over arbitrary Henselian valued fields $K$. This is a continuation of our previous article concerned with algebraic geometry over rank one valued fields. At the center of our approach is again the closedness theorem that the projections $K^{n} \times \mathbb{P}^{m}(K) \to K^{n}$ are definably closed maps. It enables application of resolution of singularities in much the same way as over locally compact ground fields. As before, the proof of that theorem uses i.a. the local behavior of definable functions of one variable and fiber shrinking, being a relaxed version of curve selection. But now, to achieve the former result, we first examine functions given by algebraic power series. All our previous results will be established here in the general settings: several versions of curve selection (via resolution of singularities) and of the \L{}ojasiewicz inequality (via two instances of quantifier elimination indicated below), extending continuous hereditarily rational functions as well as the theory of regulous functions, sets and sheaves, including Nullstellensatz and Cartan's theorems A and B. Two basic tools applied in this paper are quantifier elimination for Henselian valued fields due to Pas and relative quantifier elimination for ordered abelian groups (in a many-sorted language with imaginary auxiliary sorts) due to Cluckers--Halupczok. Other, new applications of the closedness theorem are piecewise continuity of definable functions, H\"{o}lder continuity of definable functions on closed bounded subsets of $K^{n}$, the existence of definable retractions onto closed definable subsets of $K^{n}$, and a definable, non-Archimedean version of the Tietze--Urysohn extension theorem. In a recent preprint, we established a version of the closedness theorem over Henselian valued fields with analytic structure along with some applications.Comment: This paper has been published in Journal of Singularities 21 (2020), 233-254. arXiv admin note: substantial text overlap with arXiv:1704.01093, arXiv:1703.08203, arXiv:1702.0784