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Hypergeometric periods for a tame polynomial. (English) Zbl 1113.14011
The author considers the Gauss-Manin system (GMS) of differential equations attached to a regular function satisfying a tameness assumption on a smooth affine variety over \({\mathbb C}\) (e.g. a tame polynomial on \({\mathbb C}^{n+1}\), i.e. a polynomial \(f\) for which one has \(\| \partial f\| \geq \varepsilon\) outside some compact \(K\) for some \(\varepsilon >0\)). The author considers also the Fourier transform of the GMS. He solves the Birkhoff problem and proves Hodge-type results analogous to those concerning germs of isolated hypersurface singularities.

MSC:
14D07 Variation of Hodge structures (algebro-geometric aspects)
14D05 Structure of families (Picard-Lefschetz, monodromy, etc.)
32G20 Period matrices, variation of Hodge structure; degenerations
32S40 Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects)
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