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Quantifier elimination in quasianalytic structures via non-standard analysis

Krzysztof Jan Nowak — 2015

Annales Polonici Mathematici

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....

Remarks on the generalized index of an analytic improper intersection

Krzysztof Jan Nowak — 2003

Annales Polonici Mathematici

This article continues the investigation of the analytic intersection algorithm from the perspective of deformation to the normal cone, carried out in the previous papers of the author [7, 8, 9]. The main theorem asserts that, given an analytic set V and a linear subspace S, every collection of hyperplanes, admissible with respect to an algebraic bicone B, realizes the generalized intersection index of V and S. This result is important because the conditions for a collection of hyperplanes to be...

On the Euler characteristic of the links of a set determined by smooth definable functions

Krzysztof Jan Nowak — 2008

Annales Polonici Mathematici

The purpose of this paper is to carry over to the o-minimal settings some results about the Euler characteristic of algebraic and analytic sets. Consider a polynomially bounded o-minimal structure on the field ℝ of reals. A ( C ) smooth definable function φ: U → ℝ on an open set U in ℝⁿ determines two closed subsets W := u ∈ U: φ(u) ≤ 0, Z := u ∈ U: φ(u) = 0. We shall investigate the links of the sets W and Z at the points u ∈ U, which are well defined up to a definable homeomorphism. It is proven...

A proof of the valuation property and preparation theorem

Krzysztof Jan Nowak — 2007

Annales Polonici Mathematici

The purpose of this article is to present a short model-theoretic proof of the valuation property for a polynomially bounded o-minimal theory T. The valuation property was conjectured by van den Dries, and proved for the polynomially bounded case by van den Dries-Speissegger and for the power bounded case by Tyne. Our proof uses the transfer principle for the theory T c o n v (i.e. T with an extra unary symbol denoting a proper convex subring), which-together with quantifier elimination-is due to van den...

Supplement to the paper "Quasianalytic perturbation of multi-parameter hyperbolic polynomials and symmetric matrices" (Ann. Polon. Math. 101 (2011), 275-291)

Krzysztof Jan Nowak — 2012

Annales Polonici Mathematici

In IMUJ Preprint 2009/05 we investigated the quasianalytic perturbation of hyperbolic polynomials and symmetric matrices by applying our quasianalytic version of the Abhyankar-Jung theorem from IMUJ Preprint 2009/02, whose proof relied on a theorem by Luengo on ν-quasiordinary polynomials. But those papers of ours were suspended after we had become aware that Luengo's paper contained an essential gap. This gave rise to our subsequent article on quasianalytic perturbation theory, which developed,...

Quantifier elimination, valuation property and preparation theorem in quasianalytic geometry via transformation to normal crossings

Krzysztof Jan Nowak — 2009

Annales Polonici Mathematici

This paper investigates the geometry of the expansion Q of the real field ℝ 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 Parusiński-Lion-Rolin). To this end, we study non-standard models of the universal diagram T of Q in the language ℒ augmented by the names of rational powers. Our approach makes no appeal to the Weierstrass preparation...

Division of Distributions by Locally Definable Quasianalytic Functions

Krzysztof Jan Nowak — 2010

Bulletin of the Polish Academy of Sciences. Mathematics

We demonstrate that the Łojasiewicz theorem on the division of distributions by analytic functions carries over to the case of division by quasianalytic functions locally definable in an arbitrary polynomially bounded, o-minimal structure which admits smooth cell decomposition. Hence, in particular, the principal ideal generated by a locally definable quasianalytic function is closed in the Fréchet space of smooth functions.

Intersection of analytic curves

Tadeusz KrasińskiKrzysztof Jan Nowak — 2003

Annales Polonici Mathematici

We give a relation between two theories of improper intersections, of Tworzewski and of Stückrad-Vogel, for the case of algebraic curves. Given two arbitrary quasiprojective curves V₁,V₂, the intersection cycle V₁ ∙ V₂ in the sense of Tworzewski turns out to be the rational part of the Vogel cycle v(V₁,V₂). We also give short proofs of two known effective formulae for the intersection cycle V₁ ∙ V₂ in terms of local parametrizations of the curves.

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