### DIMACS - RUTGERS EXPERIMENTAL MATHEMATICS SEMINAR

Sponsored by the Rutgers University Department of Mathematics and the

Center for Discrete Mathematics and Theoretical Computer Science (DIMACS)

**Co-organizers:**
**Brian Nakamura**, Rutgers University, bnaka {at} math [dot] rutgers [dot] edu
** Doron Zeilberger**, Rutgers University, zeilberg {at} math [dot] rutgers [dot] edu

Title: Monodromy of hypergeometric systems and analytic complexity of algebraic functions

Speaker: **Timur Sadykov**, Siberian Federal University, Krasnoyarsk

Date: Thursday, October 27, 2011 5:00pm

Location: Hill Center, Room 705, Rutgers University, Busch Campus, Piscataway, NJ

Abstract:

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.

See: http://www.math.rutgers.edu/~bnaka/expmath/