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Arbitrary rank jumps for A ‐hypergeometric systems through Laurent polynomials
Author(s) -
Matusevich Laura Felicia,
Walther Uli
Publication year - 2007
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/jdl008
Subject(s) - hypergeometric distribution , mathematics , rank (graph theory) , laurent polynomial , pure mathematics , hypergeometric function , algebra over a field , combinatorics
We investigate the solution space of hypergeometric systems of differential equations in the sense of Gel'fand, Graev, Kapranov and Zelevinskiĭ. For any integer d ⩾ 2, we construct a matrix A ( d ) ∈ ℕ d × 2 d and a parameter vector β ( d ) such that the holonomic rank of the A ‐hypergeometric systemH A ( d )( β ( 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.