Newton polytopes and algebraic hypergeometric series

Adolphson, Alan; Sperber, Steven

doi:10.1090/tran/8184

Newton polytopes and algebraic hypergeometric series
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by Alan Adolphson and Steven Sperber; with an appendix by Nicholas M. Katz PDF

Trans. Amer. Math. Soc. 373 (2020), 8365-8389 Request permission

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 $W_{2n}H_{\mathrm {DR}}^{2n}(X)$ 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.

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