We give a criterion for a real-analytic function defined on a compact nonsingular real algebraic set to be analytically equivalent to a rational function.

We study the integrals of real functions which are finite compositions of globally subanalytic maps and real power functions. These functions have finiteness properties very similar to those of subanalytic functions. Our aim is to investigate how such finiteness properties can remain when taking the integrals of such functions. The main result is that for almost all power maps arising in a $x^{\lambda }$-function, its integration leads to a non-oscillating function. This can be seen as a generalization of Varchenko...

Let $𝒥$ be a coherent ideal sheaf on a complex manifold $X$ with zero set $Z$, and let $G$ be a plurisubharmonic function such that $G=log\left|f\right|+𝒪\left(1\right)$ locally at $Z$, where $f$ is a tuple of holomorphic functions that defines $𝒥$. We give a meaning to the Monge-Ampère products ${\left(d{d}^{c}G\right)}^k$ for $k=0,1,2,...$, and prove that the Lelong numbers of the currents $M_k^{𝒥}:={\mathbf{1}}_Z{\left(d{d}^{c}G\right)}^k$ at $x$ coincide with the so-called Segre numbers of $J$ at $x$, introduced independently by Tworzewski, Gaffney-Gassler, and Achilles-Manaresi. More generally, we show that $M_k^{𝒥}$ satisfy a certain generalization...

We prove that every holomorphic bijection of a quasi-projective algebraic set onto itself is a biholomorphism. This solves the problem posed in [CR].

We construct a non-polynomially convex compact subset of the unit torus in ${ℂ}^2$ with polynomially convex hull containing no analytic structure.

Soit $f(x,y)$ une fonction sous-analytique de ${\mathbf{R}}^n\times {\mathbf{R}}^m$ à valeurs dans ${\mathbf{R}}^{+}.$ Nous montrons que l’intégrale ${\int }_{{\mathbf{R}}^m}f(x,y)dy$ est une fonction log-analytique de $x.$ Nous en déduisons que le volume $k$-dimensionnel des éléments $Y_x$ d’une famille sous-analytique de sous-ensembles sous-analytiques globaux de l’espace euclidien ${\mathbf{R}}^m$ est une fonction log-analytique de $x.$ Un corollaire de ce résultat est le caractère log-analytique de la fonction densité $k$-dimensionnelle d’un sous-analytique global de dimension $k$ en tout point de sa fermeture topologique....

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.

We give a formula for the multiplicity of a holomorphic mapping $f:{ℂ}^n\supset \Omega \to {ℂ}^m$, m > n, at an isolated zero, in terms of the degree of an analytic set at a point and the degree of a branched covering. We show that calculations of this multiplicity can be reduced to the case when m = n. We obtain an analogous result for the local Łojasiewicz exponent.

The local Nullstellensatz exponent for holomorphic mappings via intersection theory for the cases of isolated and quasi-complete intersection is considered.

We give a characterization of the irreducible components of a Weierstrass-type (W-type) analytic (resp. algebraic, Nash) variety in terms of the orbits of a Galois group associated in a natural way to this variety. Since every irreducible variety of pure dimension is (locally) a component of a W-type variety, this description may be applied to any such variety.

An effective formula for the Łojasiewicz exponent for analytic curves in a neighbourhood of 0 ∈ ℂ is given.