16 Mar 2009
My BSc thesis is titled “On the algebraicity of GKZ-hypergeometric functions defined by a (hyper)cuboid” and was written under the supervision of Frits Beukers.
My supervisor referred to this thesis in Algebraic A-hypergeometric Functions (“It has been verified by J. Schipper (…)") and my lovely girlfriend Esther Bod cited my work in Algebraicity of the Appell-Lauricella and Horn hypergeometric functions; this page exists for the benefit of those tracking down these citations.
The latter article renders this thesis mostly obsolete: it obtains the same main result in the two-dimensional case with less effort, and it leverages the idea used in the proof of the high-dimensional case to obtain a more interesting result.
GKZ-hypergeometric functions are a very general extension of hypergeometric functions. This thesis contains a nearly-complete analysis of the algebraicity of the solutions to a GKZ-system (see definition 2.1) where α is rational and A is either a rectangle, a cuboid or a hypercuboid of arbitrary dimension. It turns out that there are very few cases in which the system has algebraic solutions: see section 4.4 for details.