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Newton polytopes and algebraic hypergeometric series. (English) Zbl 1466.14024

Summary: Let \(X\) be the family of hypersurfaces in the odd-dimensional torus \(\mathbb{T}^{2n+1}\) defined by a Laurent polynomial \(f\) with fixed exponents and variable coefficients. We show that if \(n\Delta \), the dilation of the Newton polytope \(\Delta\) of \(f\) by the factor \(n\), contains no interior lattice points, then the Picard-Fuchs equation of \(W_{2n}H_{\mathrm{DR}}^{2n}(X)\) has a full set of algebraic solutions (where \(W_\bullet\) denotes the weight filtration on de Rham cohomology). We also describe a procedure for finding solutions of these Picard-Fuchs equations.

MSC:

14F40 de Rham cohomology and algebraic geometry
14D07 Variation of Hodge structures (algebro-geometric aspects)
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