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1

A remark on the intersections of subanalytic leaves

Denkowski M. P.

Opuscula Mathematica

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2016

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Vol. 36, no. 4

471--479

EN

We discuss a new sufficient condition - weaker than the usual transversality condition - for the intersection of two subanalytic leaves to be smooth. It involves the tangent cone of the intersection and, as typically non-transversal, it is of interest in analytic geometry or dynamical systems. We also prove an identity principle for real analytic manifolds and subanalytic functions.

2

Sub-Pfaffian sets and a generalization of Wilkie's theorem

Hajto Z.

Bulletin of the Polish Academy of Sciences. Mathematics

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2001

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Vol. 49, no 2

181-189

EN

We prove an analytic generalization of A. Wilkie's well-known theorem on the model completeness of the theory of the real field with exponentiation.

3

A model-theoretic version of the complement theorem : applications

Nowak K.

Bulletin of the Polish Academy of Sciences. Mathematics

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1999

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Vol. 47, no 4

355-361

EN

The paper treats of some consequences of the model-theoretic version of Gabrielov's complement theorem from [11], which asserts that the theories T[sub an] (introduced in [11] and T'[sub an] (defined herein) are model-complete. The theory T'[sub an] is a universal modification of T[sub an] in the language L'[sub an] of ordered rings expanded by the symbols of restricted analytic functions, arithmetic roots and multiplicative inverse l/x. We give a short proof of the curve selecting lemma, and next we demonstrate how quantifier elimination, within the structure R[sub an] expanded by multiplicative inverse 1/x (a result due to Denef-van den Dries [4], can be obtained from the complement theorem through a general method of logic. Also presented is an application to definability problems ; namely, a piecewise description of a subanalytic function by restricted analytic functions, arithmetic roots and l/x.

4

A model-theoretic version of the complement theorem

Nowak K.

Bulletin of the Polish Academy of Sciences. Mathematics

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1999

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Vol. 47, no 4

345-353

EN

This paper deals with an axiomatic theory T[sub an] and the expansion R[sub an] of the ordered field of reals, formed by attaching the restricted analytic functions. We show that the theory T[sub an] is model-complete, which may be regarded as a version of Gabrielov's complement theorem. Our proof is based on Robinson's test and it does not involve a partition technique. An immediate corollary is that T[sub an] coincides with the semantic theory Th(R[sub an]) of all sentences true in the structure R[sub an].

5

Borne d'invariant metrique pour une famille noetherienne

El Khadiri A.

Bulletin of the Polish Academy of Sciences. Mathematics

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1998

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Vol. 46, no 3

265-275

EN

The motivation of this paper is a question asked by B. Teissier in [1]: if [fi] : M --> N is an analytic morphism between two real analytic manifolds M and N, and if K is a compact subanalytic set of M, then for every point x[sub 0] in [fi](K) there exists an open neighbourhood U of x[sub 0] in N and a constant [gamma] > 0 such that for all x in U and all (a, b) in the same connected component of [fi^-1(x) intersection of sets K], there exist a rectifiable curve in [fi-1(x) intersection of sets K joining a and b with length less than [gamma]. In this paper we prove the following statement: let [Omega] be an open set of [R^n], N a real analytic manifold, [fi] : [Omega] --> N a proper analytic morphism and [K is a subset of set Omega] an analytic subset of [Omega]. Then for every point y[sub 0] of N there an open neighbourhood U in N and a constant [eta] > 0 such that for all y in U there exists C[sub y] > 0 satisfying the following: for every points a, b of the same connected component of [fi^-1(y) intersection of sets K] there exists an analytic rectifiable curve [sigma] in [fi^-1(y) intersection of sets] K joining a and b with [absolute value of sigma is less than or equal to] C[sub y] [absolute value a - b^eta], where [absolute value of sigma] is the length of [sigma] and [absolute value a - b] is the euclidean distance between a and b.