Newton polytopes and algebraic hypergeometric series

Alan Adolphson; Steven Sperber; Nicholas M. Katz

doi:10.1090/tran/8184

Newton polytopes and algebraic hypergeometric series

Alan Adolphson, Steven Sperber, Nicholas M. Katz

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

Let X be the family of hypersurfaces in the odd-dimensional torus T²n⁺¹ defined by a Laurent polynomial f with fixed exponents and variable coefficients. We show that if nΔ, the dilation of the Newton polytope Δ of f by the factor n, contains no interior lattice points, then the Picard-Fuchs equation of W_2nH_DR²ⁿ (X) has a full set of algebraic solutions (where W_• denotes the weight filtration on de Rham cohomology). We also describe a procedure for finding solutions of these Picard-Fuchs equations.

Original language	English (US)
Pages (from-to)	8365-8389
Number of pages	25
Journal	Transactions of the American Mathematical Society
Volume	373
Issue number	12
DOIs	https://doi.org/10.1090/tran/8184
State	Published - Dec 2020

All Science Journal Classification (ASJC) codes

General Mathematics
Applied Mathematics

Keywords

Hypergeometric series
Newton polytope
Picard-Fuchs equation

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10.1090/tran/8184

Cite this

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AU - Adolphson, Alan

AU - Sperber, Steven

AU - Katz, Nicholas M.

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N2 - Let X be the family of hypersurfaces in the odd-dimensional torus T2n+1 defined by a Laurent polynomial f with fixed exponents and variable coefficients. We show that if nΔ, the dilation of the Newton polytope Δ of f by the factor n, contains no interior lattice points, then the Picard-Fuchs equation of W2nHDR2n (X) has a full set of algebraic solutions (where W• denotes the weight filtration on de Rham cohomology). We also describe a procedure for finding solutions of these Picard-Fuchs equations.

AB - Let X be the family of hypersurfaces in the odd-dimensional torus T2n+1 defined by a Laurent polynomial f with fixed exponents and variable coefficients. We show that if nΔ, the dilation of the Newton polytope Δ of f by the factor n, contains no interior lattice points, then the Picard-Fuchs equation of W2nHDR2n (X) has a full set of algebraic solutions (where W• denotes the weight filtration on de Rham cohomology). We also describe a procedure for finding solutions of these Picard-Fuchs equations.

KW - Hypergeometric series

KW - Newton polytope

KW - Picard-Fuchs equation

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JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

IS - 12

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Newton polytopes and algebraic hypergeometric series

Abstract

All Science Journal Classification (ASJC) codes

Keywords

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