## Kavli IPMU Komaba Seminar

Seminar information archive ～04/01｜Next seminar｜Future seminars 04/02～

Date, time & place | Monday 16:30 - 18:00 002Room #002 (Graduate School of Math. Sci. Bldg.) |
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### 2010/02/01

16:30-18:00 Room #002 (Graduate School of Math. Sci. Bldg.)

Bases in the solution space of the Mellin system

**Timur Sadykov**(Siberian Federal University)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

\\begin{equation}

a_0 z^m + a_1z^{m_1} + a_2 z^{m_2} + \\ldots + a_n z^{m_n} +

a_{n+1} =0.

\\end{equation}

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.

I will present a joint work with Alicia Dickenstein.

We consider algebraic functions $z$ satisfying equations of the

form

\\begin{equation}

a_0 z^m + a_1z^{m_1} + a_2 z^{m_2} + \\ldots + a_n z^{m_n} +

a_{n+1} =0.

\\end{equation}

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.