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Mathematics 2017, 5(4), 52; https://doi.org/10.3390/math5040052

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

Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599-3250, USA
Received: 22 September 2017 / Revised: 13 October 2017 / Accepted: 13 October 2017 / Published: 17 October 2017
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Abstract

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. View Full-Text
Keywords: Gauss–Manin differential equations; multidimensional hypergeometric integrals; Bethe ansatz Gauss–Manin differential equations; multidimensional hypergeometric integrals; Bethe ansatz
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
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Varchenko, A. Solutions Modulo p of Gauss–Manin Differential Equations for Multidimensional Hypergeometric Integrals and Associated Bethe Ansatz. Mathematics 2017, 5, 52.

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