## Kavli IPMU Komaba Seminar

Date, time & place Monday 16:30 - 18:00 002Room #002 (Graduate School of Math. Sci. Bldg.)

### 2010/02/01

16:30-18:00   Room #002 (Graduate School of Math. Sci. Bldg.)
Bases in the solution space of the Mellin system
[ Abstract ]
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.