Monodromy of hypergeometric systems and analytic complexity of algebraic functions

A system of partial differential equations of hypergeometric type can be determined by specifying an integer matrix of maximal rank together with a complex vector of parameters. We will say that such a system of equations has maximally reducible monodromy if its space of local holomorphic solutions in a neighbourhood of a generic point splits into the direct sum of one-dimensional invariant subspaces. In the talk, I will present necessary and sufficient conditions for the monodromy of a bivariate nonconfluent hypergeometric system to be maximally reducible. In particular, any bivariate system defined by a matrix whose rows determine a plane zonotope, admits maximally reducible monodromy for some choice of the vector of its complex parameters.
As an application, I will deduce estimates on the analytic complexity of bivariate algebraic functions. According to V. K. Beloshapka's definition, the order of complexity of any univariate function is equal to zero while the $n$-th complexity class is defined recursively to consist of functions of the form $a(b(x,y)+c(x,y))$, where $a$ is a univariate analytic function and $b$ and $c$ belong to the $(n-1)$-th complexity class. Such a represenation is meant to be valid for suitable germs of multi-valued holomorphic functions.
A randomly chosen bivariate analytic functions will most likely have infinite analytic complexity. However, for a number of important families of algebraic functions their complexity is finite and can be computed or estimated. Using properties of solutions to the Gelfand–Kapranov–Zelevinsky system we obtain estimates for the analytic complexity of such functions.