# IPMU Komaba seminar

Speaker: Timur Sadykov (Siberian Federal University) Bases in the solution space of the Mellin system Mon, Feb 01, 2010, 16:30 - 18:00 Room 002, Mathematical Sciences Building, Komaba Campus I will present a joint work with Alicia Dickenstein. We consider algebraic functions $z$ satisfying equations of the form $$a_0 z^m + a_1z^{m_1} + a_2 z^{m_2} + \ldots + a_n z^{m_n} + a_{n+1} =0.$$ Here $m > m_1 > \ldots > m_n>0,$ $m,m_i \in \N,$ and $z=z(a_0,\ldots,a_{n+1})$ is a function of the complex variables $a_0, \ldots, a_{n+1}.$ Solutions to such equations are classically known to satisfy holonomic systems of linear partial differential equations with polynomial coefficients. In the talk I will investigate one of such systems of differential equations which was introduced by Mellin. We compute the holonomic rank of the Mellin system as well as the dimension of the space of its algebraic solutions. Moreover, we construct explicit bases of solutions in terms of the roots of initial algebraic equation and their logarithms. We show that the monodromy of the Mellin system is always reducible and give some factorization results in the univariate case.