Newton polytopes and algebraic hypergeometric series

Let $X$ be the family of hypersurfaces in the odd-dimensional torus ${\mathbb T}^{2n+1}$definedbyaLaurentpolynomial$f$ with fixed exponents and variable coefficients. We show that if $n\mathrm{\Delta }$, the dilation of the Newton polytope $\mathrm{\Delta }$ of $f$ by the factor $n$, contains no interior lattice points, then the Picard-Fuchs equation of $W_{2n}{H}_{\mathrm{D}\mathrm{R}}^{2n}(X)$ 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.