A classification of the irreducible algebraic A-hypergeometric functions associated to planar point configurations

TL;DR

This paper classifies planar point configurations that produce algebraic A-hypergeometric functions, showing that irreducible algebraic functions are rare when boundary points are large and A is not a pyramid.

Contribution

It provides a complete classification of point configurations in the plane leading to algebraic A-hypergeometric functions, highlighting conditions for their existence.

Findings

No irreducible algebraic functions with many boundary points unless A is a pyramid.

Identifies all configurations with algebraic hypergeometric functions.

Establishes conditions under which algebraic solutions exist.

Abstract

We consider A-hypergeometric functions associated to normal sets in the plane. We give a classification of all point configurations for which there exists a parameter vector such that the associated hypergeometric function is algebraic. In particular, we show that there are no irreducible algebraic functions if the number of boundary points is sufficiently large and A is not a pyramid.