Let X be the family of hypersurfaces in the odd-dimensional torus T2n+1 defined by a Laurent polynomial f with fixed exponents and variable coefficients. We show that if nΔ, the dilation of the Newton polytope Δ of f by the factor n, contains no interior lattice points, then the Picard-Fuchs equation of W2nHDR2n (X) has a full set of algebraic solutions (where W• denotes the weight filtration on de Rham cohomology). We also describe a procedure for finding solutions of these Picard-Fuchs equations.
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- Hypergeometric series
- Newton polytope
- Picard-Fuchs equation