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Expansion of the solutions of a Gauss-Manin system at a point of infinity. (English) Zbl 0554.32009
If f(x) is a polynomial of n complex variables, $$x_ 1,...,x_ n$$, let $$F_ 0(t,x)$$ be a deformation of f. We set $$F=t_ 0+F_ 0$$ and consider integrals of the form $$(*)\quad u=\int \delta^{(\lambda)}(F)dx$$ or $$(**)\quad u=\int F^{-\lambda -1}dx$$ where $$dx=dx_ 1\wedge...\wedge dx_ n$$ and $$\lambda\in {\mathbb{C}}$$. Formulas (*) or (**) form a so-called ”Gauss-Manin” system either of which defines a meromorphic connection on $$S=\{(t_ 0,t)\}$$. This connection has poles which lie in the discriminant variety D. Thus the author studies many-valued, holomorphic solutions of the Gauss-Manin system in $$S\setminus D$$. He obtains power series representations for these solutions.
Reviewer: S.G.Krantz

##### MSC:
 32C30 Integration on analytic sets and spaces, currents 58H15 Deformations of general structures on manifolds 53C05 Connections, general theory 32S30 Deformations of complex singularities; vanishing cycles
##### Keywords:
deformation of polynomial; meromorphic connection
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