Newton polytopes and algebraic hypergeometric series

TL;DR

This paper investigates the conditions under which the Picard-Fuchs equations for certain hypersurfaces in odd-dimensional tori have algebraic solutions, linking geometric properties of Newton polytopes to differential equations.

Contribution

It establishes a criterion involving Newton polytopes for the existence of algebraic solutions to Picard-Fuchs equations of hypersurfaces in tori and provides a method to find these solutions.

Findings

Full set of algebraic solutions when $n riangle$ contains no interior lattice points

Criterion linking Newton polytope dilation to algebraic solutions

Procedure for explicitly finding solutions to Picard-Fuchs equations

Abstract

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.