  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 $W_{2n}H^{2n}_{\rm DR}(X)$ 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.
