Newton polytopes and algebraic hypergeometric series.

Let X be the family of hypersurfaces in the odd-dimensional torus T²ⁿ⁺¹ defined by a Laurent polynomial ƒ with fixed exponents and variable coefficients. We show that if n Δ, the dilation of the Newton polytope Δ of ƒ by the factor n, contains no interior lattice points, then the Picard-Fuchs equation of W_2nH_DR²ⁿ(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. [ABSTRACT FROM AUTHOR]