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1

On intersections with hypersurfaces and normal pseudo-flatness

Rams S.

Bulletin of the Polish Academy of Sciences. Mathematics

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2000

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Vol. 48, no 3

335-340

EN

We study the behaviour of the families of hypersurfaces that are applied in [4] to estimate the Łojasiewicz exponent and the exponent of separation at infinity. We prove that the hypersurfaces for which v((Z . H)^S, c) is minimal are to be smooth along a subset of Z [intersection of sets] S and to meet some algebraic cones along the tangent space to S.

2

Invariance of generalized L-regularity under holomorphic mappings

Hecart J.

Bulletin of the Polish Academy of Sciences. Mathematics

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2000

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Vol. 48, no 2

187-192

EN

We show an invariance property for generalized L-regular sets (that is regular sets with respect to the generalized pluricomplex Green function defined in parabolic complex spaces) under holomorphic mappings.

3

On the Łojasiewicz exponent of the gradient of holomorphic functions

Bogusławska M.

Bulletin of the Polish Academy of Sciences. Mathematics

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1999

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Vol. 47, no 4

337-343

EN

We show that the Łojasiewicz exponent L[sub o] (grad f) at zero of the gradient of a distinguished pseudo-polynomial f [...] is attained on the zero-set of f'[sub y].

4

Lemmass A and B for sub-Pfaffian sets

Hajto Z.

Bulletin of the Polish Academy of Sciences. Mathematics

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1999

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Vol. 47, no 4

325-336

EN

In this paper we prove two fundamental lemmas of sub-Pfaffian geometry which are counterparts of Lemmas A and B for subanalytic sets [4]. We use a generalized version of the Tangent Mapping Theorem [2], following our program announced in [11].

5

Descriptive properties of spaces of measures

Holicky P., Kalenda O.

Bulletin of the Polish Academy of Sciences. Mathematics

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1999

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Vol. 47, no 1

37-51

EN

The spaces of Borel probabilities on a topological space X inherit a number of topological properties of X. We show in particular that the space of tight probabilities on a Cech-analytic space is Cech-analytic. Analogical results are shown for several other classes of generalized analytic and complete topological spaces.

6

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.

7

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].

8

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.

9

Bezout-type theorems for certain analytic sets

Rams S.

Bulletin of the Polish Academy of Sciences. Mathematics

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1998

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Vol. 46, no 3

277-283

EN

We give a formula for the total number of intersections of certain analytic subsets of cartesian products of holomorphically separable second-countable complex manifolds (Theorems 4.1, 4.2). We prove some stronger results for curves in cartesian products of open subsets of C (Theorem 4.3).