## Abstract

The rank of an (Formula presented.) -hypergeometric (Formula presented.) -module (Formula presented.), associated with a full-rank (Formula presented.) -matrix (Formula presented.) and a vector of parameters (Formula presented.), is known to be the normalized volume of (Formula presented.), denoted (Formula presented.), when (Formula presented.) lies outside the exceptional arrangement (Formula presented.), an affine subspace arrangement of codimension at least two. If (Formula presented.) is simple, we prove that (Formula presented.) is a tight upper bound for the ratio (Formula presented.) for any (Formula presented.). We also prove that the set of parameters (Formula presented.) such that this ratio is at least two is an affine subspace arrangement of codimension at least three.

Original language | English (US) |
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Pages (from-to) | 182-192 |

Number of pages | 11 |

Journal | Bulletin of the London Mathematical Society |

Volume | 54 |

Issue number | 1 |

DOIs | |

State | Published - Feb 2022 |

### Bibliographical note

Funding Information:We are grateful to Laura Felicia Matusevich and Uli Walther for helpful discussions over the years on bounding the rank of an ‐hypergeometric system. C. Berkesch was partially supported by NSF Grant DMS 1661962 and NSF Grant DMS 2001101. M.‐C. Fernández‐Fernández was partially supported by MTM2016‐75024‐P and FEDER.

Funding Information:

We are grateful to Laura Felicia Matusevich and Uli Walther for helpful discussions over the years on bounding the rank of an A$A$-hypergeometric system. C. Berkesch was partially supported by NSF Grant DMS 1661962 and NSF Grant DMS 2001101. M.-C. Fern?ndez-Fern?ndez was partially supported by MTM2016-75024-P and?FEDER.

Publisher Copyright:

© 2022 The Authors. The publishing rights in this article are licensed to the London Mathematical Society under an exclusive licence.