A sharp bound for hypergeometric rank in dimension three

TL;DR

This paper establishes a precise upper limit of two for the ratio of the rank of a three-dimensional A-hypergeometric system to its normalized volume, refining understanding of these systems' complexity.

Contribution

It introduces a sharp upper bound for the hypergeometric rank ratio specifically in three-dimensional cases, improving previous bounds.

Findings

The ratio of hypergeometric rank to normalized volume is at most two in three dimensions.

The bound is proven to be sharp, meaning it is the best possible.

Provides new insights into the structure of A-hypergeometric systems in three dimensions.

Abstract

We provide a sharp upper bound on the quotient of the rank of an A-hypergeometric system with a three-dimensional torus action by the normalized volume of A; in this case, the upper bound is two.