Solutions Modulo p of Gauss–Manin Differential Equations for Multidimensional Hypergeometric Integrals and Associated Bethe Ansatz

We consider the Gauss–Manin differential equations for hypergeometric integrals associated with a family of weighted arrangements of hyperplanes moving parallel to themselves. We reduce these equations modulo a prime integer p and construct polynomial solutions of the new differential equations as p...

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Bibliographic Details
Published in:Mathematics (Basel) Vol. 5; no. 4; p. 52
Main Author: Varchenko, Alexander
Format: Journal Article
Language:English
Published: Basel MDPI AG 17.10.2017
Subjects:
Analogs
Bethe ansatz
Differential equations
Elliptic functions
Food science
Gauss–Manin differential equations
Geometry
Hyperplanes
Integrals
Mathematical analysis
multidimensional hypergeometric integrals
ISSN:2227-7390, 2227-7390
Online Access:Get full text
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Summary:We consider the Gauss–Manin differential equations for hypergeometric integrals associated with a family of weighted arrangements of hyperplanes moving parallel to themselves. We reduce these equations modulo a prime integer p and construct polynomial solutions of the new differential equations as p-analogs of the initial hypergeometric integrals. In some cases, we interpret the p-analogs of the hypergeometric integrals as sums over points of hypersurfaces defined over the finite field Fp. This interpretation is similar to the classical interpretation by Yu. I. Manin of the number of points on an elliptic curve depending on a parameter as a solution of a Gauss hypergeometric differential equation. We discuss the associated Bethe ansatz.
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ISSN:2227-7390
2227-7390
DOI:10.3390/math5040052