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Journal of the London Mathematical Society Advance Access originally published online on January 31, 2007
Journal of the London Mathematical Society 2007 75(1):213-224; doi:10.1112/jlms/jdl008
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© 2007 London Mathematical Society

Arbitrary rank jumps for A-hypergeometric systems through Laurent polynomials

Laura Felicia Matusevich

Department of Mathematics
Harvard University
Cambridge
Harvard 02138
MA
USA
laura{at}math.tamu.edu

Uli Walther

Department of Mathematics
Purdue University
West Lafayette
Purdue 47907
IN
USA
walther{at}math.purdue.edu

We investigate the solution space of hypergeometric systems of differential equations in the sense of Gel’fand, Graev, Kapranov and Zelevinskii. For any integer d ≥ 2, we construct a matrix A(d) isin Nd x 2d and a parameter vector ß(d) such that the holonomic rank of the A-hypergeometric system HA(d)(ß(d)) exceeds the simplicial volume vol(A(d)) by at least d – 1. The largest previously known gap between rank and volume was 2.

Our construction gives evidence to the general observation that rank jumps seem to go hand in hand with the existence of multiple Laurent (or Puiseux) polynomial solutions.

Current address: Department of Mathematics, Texas A&M University, College Station TX 77843, USA

1991 Mathematics Subject Classification 13N10, 33C70, 14M25, 52B20.

The first author was partially supported by an NSF Postdoctoral Fellowship. The second author was partially supported by the NSF, the DfG, the Humboldt Foundation, and the NSA.

Received April 16, 2004; published online January 31, 2007.


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