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1 Neighborhoods 2 1.1 Preliminary Notations and Definitions . . . . . . . . . . . . . 3 1.2 Formal Principle . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3 Formal Principle for Singularities . . . . . . . . . . . . . . . . 8 1.4 Pseudoconvex Domains . . . . . . . . . . . . . . . . . . . . . . 10 1.5 Exceptional Sets . . . . . . . . . . . . . . . .… (More)

We study the analogue of the infinitesimal 16th Hilbert problem in dimension zero. Lower and upper bounds for the number of the zeros of the corresponding Abelian integrals (which are algebraic functions) are found. We study the relation between the vanishing of an Abelian integral I(t) defined over Q and its arithmetic properties. Finally, we give… (More)

- Hossein Movasati
- 2004

The aim of this paper is to introduce the theory of Abelian integrals for holomorphic foliations in a complex manifold of dimension two. We will show the importance of Picard-Lefschetz theory and the classification of relatively exact 1-forms in this theory. As an application we identify some irreducible components of the space of holomorphic foliations of… (More)

- Hossein Movasati
- 2002

In this paper we prove that any degree d deformation of a generic logarithmic polynomial differential equation with a persistent center must be logarithmic again. This is a generalization of Ilyashenko’s result on Hamiltonian differential equations. The main tools are PicardLefschetz theory of a polynomial with complex coefficients in two variables,… (More)

- Hossein Movasati
- 2002

The main objective of this article is to study the topology of the fibers of a generic rational function of the type F p Gq in the projective space of dimension two. We will prove that the action of the monodromy group on a single Lefschetz vanishing cycle δ generates the first homology group of a generic fiber of F p Gq . In particular, we will prove that… (More)

- Hossein Movasati
- 2005

In this article we introduce algorithms which compute iterations of Gauss-Manin connections, Picard-Fuchs equations of Abelian integrals and mixed Hodge structure of affine varieties of dimension n in terms of differential forms. In the case n = 1 such computations have many applications in differential equations and counting their limit cycles. For n > 3,… (More)

- Hossein Movasati
- 2006

Abstract In the present article we define the algebra of differential modular forms and we prove that it is generated by Eisenstein series of weight 2, 4 and 6. We define Hecke operators on them, find some analytic relations between these Eisenstein series and obtain them in a natural way as coefficients of a family of elliptic curves. The fact that a… (More)

In this article we study deformations of a holomorphic foliation with a generic non-rational first integral in the complex plane. We consider two vanishing cycles in a regular fiber of the first integral with a non-zero self intersection and with vanishing paths which intersect each other only at their start points. It is proved that if the deformed… (More)

- Hossein Movasati
- 2004

In this article we introduce the mixed Hodge structure of the Brieskorn module of a polynomial f in C, where f satisfies a certain regularity condition at infinity (and hence has isolated singularities). We give an algorithm which produces a basis of a localization of the Brieskorn module which is compatible with its mixed Hodge structure. As an application… (More)

- Hossein Movasati
- 2006

In this article we give an algorithm which produces a basis of the n-th de Rham cohomology of the affine smooth hypersurface f−1(t) compatible with the mixed Hodge structure, where f is a polynomial in n+ 1 variables and satisfies a certain regularity condition at infinity (and hence has isolated singularities). As an application we show that the notion of… (More)