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arXiv · 0807.0453

The rank of a hypergeometric system

Christine Berkesch
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Abstract

The holonomic rank of the A-hypergeometric system M_A(β) is the degree of the toric ideal I_A for generic parameters; in general, this is only a lower bound. To the semigroup ring of A we attach the ranking arrangement and use this algebraic invariant and the exceptional arrangement of nongeneric parameters to construct a combinatorial formula for the rank jump of M_A(β). As consequences, we obtain a refinement of the stratification of the exceptional arrangement by the rank of M_A(β) and show that the Zariski closure of each of its strata is a union of translates of linear subspaces of the parameter space. These results hold for generalized A-hypergeometric systems as well, where the semigroup ring of A is replaced by a nontrivial weakly toric module M contained in \CC[\ZZ A]. We also provide a direct proof of the result of M. Saito and W. Traves regarding the isomorphism classes of M_A(β).

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Christine Berkesch. 2010-02-10. The rank of a hypergeometric system. https://doi.org/10.1112/s0010437x10004811

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