Integral ratios of factorials and algebraic hypergeometric functions

TL;DR

This paper explores the relationship between integral ratios of factorials and algebraic hypergeometric functions, extending classical results and connecting monodromy groups with quadratic forms.

Contribution

It provides a proof sketch linking factorial ratios and algebraic hypergeometric functions, utilizing Beukers and Heckman's characterization and Bezoutian constructions.

Findings

Extension of Landau's results to algebraic hypergeometric functions

Identification of Hermitian forms fixed by monodromy groups

Connection between Bezoutian quadratic forms and monodromy

Abstract

Sketch of proof of a theorem relating the two subjects of the title. It can be thought as an extension of results of Landau for the classical hypergeometric function. It relies on the characterization of algebraic hypergeometric functions of Beukers and Heckman. In the process we also show that a variant of a classical construction of Bezout (producing a quadratic form, the Bezoutian, out of two polynomials in one variable) gives the Hermitian form fixed by the monodromy group, up to scaling.