Nilsson solutions for irregular A-hypergeometric systems

TL;DR

This paper investigates irregular A-hypergeometric systems, constructing solutions as Nilsson series via Gr"obner degenerations, and establishes conditions for convergence and the nature of singularities.

Contribution

It provides explicit constructions of solutions as Nilsson series for irregular A-hypergeometric systems and offers an alternative proof regarding their irregular singularities.

Findings

Solutions are formal Nilsson series constructed from Gr"obner degenerations.

Convergence of series when the weight vector is a perturbation of (1,...,1).

Irregular singularities are present in inhomogeneous A-hypergeometric systems.

Abstract

We study the solutions of irregular A-hypergeometric systems that are constructed from Gr\"obner degenerations with respect to generic positive weight vectors. These are formal logarithmic Puiseux series that belong to explicitly described Nilsson rings, and are therefore called (formal) Nilsson series. When the weight vector is a perturbation of (1,...,1), these series converge and provide a basis for the (multivalued) holomorphic hypergeometric functions in a specific open subset of complex n-space. Our results are more explicit when the parameters are generic or when the solutions studied are logarithm-free. We also give an alternative proof of a result of Schulze and Walther that inhomogeneous A-hypergeometric systems have irregular singularities.