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arXiv · 1806.10243

Newton polytopes and algebraic hypergeometric series

Alan Adolphson·Steven Sperber
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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.

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Alan Adolphson, Steven Sperber. 2018-06-26. Newton polytopes and algebraic hypergeometric series. https://arxiv.org/abs/1806.10243

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