A note on Bézout's theorem
We present a version of Bézout's theorem basing on the intersection theory in complex analytic geometry. Some applications for products of surfaces and curves are also given.
A Theorem on Solutions of Analytic Equations with Applications to Deformations of Complex Structures.
Complex Analytic and Formal Solutions of Real Analytic Equations in Cn.
Constructions des ensembles déterminants pour les fonctions analytiques
Continuity of total number of intersection
Coverings defined by Weierstraß polynomials.
Eine bemerkenswerte Eigenschaft der formellen Fasern affinoider Räume.
Eine Linearisierung kontrahierender biholomorpher Abbildungen und damit zusammenhängender analytischer Differentialgleichungssysteme.
Flatness testing over singular bases
We show that non-flatness of a morphism φ:X→ Y of complex-analytic spaces with a locally irreducible target of dimension n manifests in the existence of vertical components in the n-fold fibred power of the pull-back of φ to the desingularization of Y. An algebraic analogue follows: Let R be a locally (analytically) irreducible finite type ℂ-algebra and an integral domain of Krull dimension n, and let S be a regular n-dimensional algebra of finite type over R (but not necessarily a finite R-module),...
Holomorphic non-holonomic differential systems on complex manifolds
We study coherent subsheaves 𝓓 of the holomorphic tangent sheaf of a complex manifold. A description of the corresponding 𝓓-stable ideals and their closed complex subspaces is sketched. Our study of non-holonomicity is based on the Noetherian property of coherent analytic sheaves. This is inspired by the paper [3] which is related with some problems of mechanics.
Hypersurfaces intégrales des feuilletages holomorphes
Soit un germe en de 1-forme différentielle holomorphe, satisfaisant la condition d’intégrabilité et non dicritique, i.e. sur toute surface non intégrale de , on ne peut tracer, au voisinage de 0, qu’un nombre fini de germes de courbes analytiques , intégrales de , avec . Alors possède un germe d’hypersurface analytique intégrale.
Le lemme fondamental de Nilsson dans le cas analytique local
On donne des évaluations précises de la croissance modérée des intégrales de fonctions de classe de Nilsson locale dans , exprimées par des caractéristiques topologiques des courbes de ramification des intégrands.
Local Complex Foliation of Real Submanifolds.
Local geometry of zero sets of holomorphic functions near the torus.
...-Modules on Supermanifolds.
Multiplicity and the Lojasiewicz exponent
On gradient at infinity of semialgebraic functions
Let f: ℝⁿ → ℝ be a C² semialgebraic function and let c be an asymptotic critical value of f. We prove that there exists a smallest rational number such that |x|·|∇f| and are separated at infinity. If c is a regular value and , then f is a locally trivial fibration over c, and the trivialisation is realised by the flow of the gradient field of f.
On quasi-free Hilbert modules.
On the Łojasiewicz exponent for analytic curves
An effective formula for the Łojasiewicz exponent for analytic curves in a neighbourhood of 0 ∈ ℂ is given.
On the Łojasiewicz exponent of the gradient of a holomorphic function
The Łojasiewicz exponent of the gradient of a convergent power series h(X,Y) with complex coefficients is the greatest lower bound of the set of λ > 0 such that the inequality holds near for a certain c > 0. In the paper, we give an estimate of the Łojasiewicz exponent of grad h using information from the Newton diagram of h. We obtain the exact value of the exponent for non-degenerate series.