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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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 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.",

keywords = "Hypergeometric series, Newton polytope, Picard-Fuchs equation",

author = "Alan Adolphson and Steven Sperber and Katz, \{Nicholas M.\}",

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year = "2020",

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doi = "10.1090/tran/8184",

language = "English (US)",

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T1 - Newton polytopes and algebraic hypergeometric series

AU - Adolphson, Alan

AU - Sperber, Steven

AU - Katz, Nicholas M.

PY - 2020/12

Y1 - 2020/12

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

UR - https://www.scopus.com/pages/publications/85096641321

UR - https://www.scopus.com/pages/publications/85096641321#tab=citedBy

U2 - 10.1090/tran/8184

DO - 10.1090/tran/8184

M3 - Article

AN - SCOPUS:85096641321

SN - 0002-9947

VL - 373

SP - 8365

EP - 8389

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

IS - 12

ER -

Newton polytopes and algebraic hypergeometric series

Abstract

All Science Journal Classification (ASJC) codes

Keywords

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