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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
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