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$ {A}$-hypergeometric systems that come from geometry


Authors: Alan Adolphson and Steven Sperber
Journal: Proc. Amer. Math. Soc. 140 (2012), 2033-2042
MSC (2010): Primary 33C70, 14F40; Secondary 52B20
DOI: https://doi.org/10.1090/S0002-9939-2011-11073-6
Published electronically: October 13, 2011
MathSciNet review: 2888191
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Abstract: In recent work, Beukers characterized $ {A}$-hypergeometric systems having a full set of algebraic solutions. He accomplished this by (1) determining which $ {A}$-hypergeometric systems have a full set of polynomial solutions modulo $ p$ for almost all primes $ p$ and (2) showing that these systems come from geometry. He then applied a fundamental theorem of N. Katz, which says that such systems have a full set of algebraic solutions. In this paper we establish some connections between nonresonant $ A$-hypergeometric systems and de Rham-type complexes, which leads to a determination of which $ A$-hypergeometric systems come from geometry. We do not use the fact that the system is irreducible or find integral formulas for its solutions.


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Additional Information

Alan Adolphson
Affiliation: Department of Mathematics, Oklahoma State University, Stillwater, Oklahoma 74078
Email: adolphs@math.okstate.edu

Steven Sperber
Affiliation: School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455
Email: sperber@math.umn.edu

DOI: https://doi.org/10.1090/S0002-9939-2011-11073-6
Keywords: $A$-hypergeometric system, de Rham cohomology
Received by editor(s): December 9, 2010
Received by editor(s) in revised form: January 24, 2011, and February 9, 2011
Published electronically: October 13, 2011
Communicated by: Lev Borisov
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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