-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

Published electronically:
October 13, 2011

MathSciNet review:
2888191

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Abstract | References | Similar Articles | Additional Information

Abstract: In recent work, Beukers characterized -hypergeometric systems having a full set of algebraic solutions. He accomplished this by (1) determining which -hypergeometric systems have a full set of polynomial solutions modulo for almost all primes 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 -hypergeometric systems and de Rham-type complexes, which leads to a determination of which -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.