On the $b$-functions of hypergeometric systems

TL;DR

This paper investigates the roots of $b$-functions associated with $A$-hypergeometric systems, providing bounds and estimates for their behavior along coordinate hyperplanes and at generic points.

Contribution

It offers new bounds and estimates for the roots of $b$-functions of hypergeometric systems and their Fourier transforms, advancing understanding of their singularity structure.

Findings

Bounds for roots of $b$-functions along hyperplanes

Estimates for $b$-functions at generic points

Analysis of $b$-functions of Fourier transforms of hypergeometric systems

Abstract

For any integer $d\times (n+1)$ matrix $A$ and parameter $\beta\in\CC^d$ let $M_A(\beta)$ be the associated $A$-hypergeometric (or GKZ) system in the variables $x_0,\ldots,x_n$. We describe bounds for the (roots of the) $b$-functions of both $M_A(\beta)$ and its Fourier transform along the hyperplanes $(x_j=0)$. We also give an estimate for the $b$-function for restricting $M_A(\beta)$ to a generic point.