Pfaffian Systems of A-Hypergeometric Equations I: Bases of Twisted Cohomology Groups

TL;DR

This paper investigates bases of Pfaffian systems for A-hypergeometric equations, linking algebraic, combinatorial, and geometric methods to describe and analyze these bases, especially for specific classes of polytopes and posets.

Contribution

It introduces a method to construct bases of Pfaffian systems via Gröbner deformations and provides combinatorial descriptions for certain hypergeometric systems.

Findings

Bases can be constructed using Gröbner deformations.

Combinatorial descriptions are available for systems related to order polytopes.

Growth rate of bases for certain subclasses is polynomial.

Abstract

This is the third revision. We study bases of Pfaffian systems for $A$-hypergeometric system. Gr\"obner deformations give bases. These bases also give those for twisted cohomology groups. For hypergeometric system associated to a class of order polytopes, these bases have a combinatorial description. The size of the bases associated to a subclass of the order polytopes have the growth rate of the polynomial order. Bases associated to two chain posets and bouquets are studied.