## Classical Analysis

Seminar information archive ～05/22｜Next seminar｜Future seminars 05/23～

### 2012/07/18

16:00-17:30 Room #128 (Graduate School of Math. Sci. Bldg.)

Free divisors, holonomic systems and algebraic Painlev\\'{e} sixth solutions (ENGLISH)

**Jiro Sekiguchi**(Tokyo University of Agriculture and Technology)Free divisors, holonomic systems and algebraic Painlev\\'{e} sixth solutions (ENGLISH)

[ Abstract ]

In this talk, I will report an attempt to treat algebraic solutions of Painlev\\'{e} VI equation in a unified manner.

A classification of algebraic solutions of Painlev\\'{e} VI equation was accomplished by O. Lisovyy and Y. Tykhyy after efforts on the construction of such solutions by many authors, K. Iwasaki N. J. Hitchin, P. Boalch, B. Dubrovin, M. Mazzocco, A. V. Kitaev, R. Vidunas and others.

The outline of my approach is as follows.

Let $t$ be a variable and let $w$ be its algebraic function such that $w$ is a solution of Painlev\\'{e} sixth equation. Suppose that both $t$ and $w$ are rational functions of a parameter. Namely $(t,w)$ defines a rational curve.

(1) Find a polynomial $P(u)$ such that $t=\\frac{P(-u)}{P(u)}$.

(2) From $P(u)$, define a weighted homogeneous polynomial $f(x_1,x_2,x_3)=x_3f_1(x_1,x_2,x_3)$ of three variables $x_1,x_2,x_3$, where $(1,2,n)$ is the weight system of $(x_1,x_2,x_3)$ with $n=\\deg P(u)$. The hypersurface $D:f(x)=0$ is a free divisor in ${\\bf C}^3$. Note that $\\deg_{x_3}f_1=2$.

(3) Construct a holonomic system ${\\sl M}$ on ${\\bf C}^3$ of rank two with singularities along $D$.

(4) Construct an ordinary differential equation from the holonomic system ${\\sl M}$ with respect to $x_3$. This differential equation has three singular points $z_0,z_1,a_s$ in $x_3$-line.

(5) Putting $t=\\frac{z_1}{z_0},\\lambda=\\frac{a_s}{z_0}$, we conclude that $(t,\\lambda)$ is equivalent to the pair $(t,w)$.

Our study starts with showing the existence of $P(u)$ in (1). From the classification by Losovyy and Tykhyy, I find that the existence of $P(u)$ is guaranteed for Solutions III, IV, Solutions $k$ ($1\\le k\\le 21$, $k\\not= 4,13,14,20$) and Solution 30. We checked whether (1)-(5) are true or not in these cases separately and as a consequence (1)-(5) hold for the all these cases except Solutions 19, 21.

In this talk, I will report an attempt to treat algebraic solutions of Painlev\\'{e} VI equation in a unified manner.

A classification of algebraic solutions of Painlev\\'{e} VI equation was accomplished by O. Lisovyy and Y. Tykhyy after efforts on the construction of such solutions by many authors, K. Iwasaki N. J. Hitchin, P. Boalch, B. Dubrovin, M. Mazzocco, A. V. Kitaev, R. Vidunas and others.

The outline of my approach is as follows.

Let $t$ be a variable and let $w$ be its algebraic function such that $w$ is a solution of Painlev\\'{e} sixth equation. Suppose that both $t$ and $w$ are rational functions of a parameter. Namely $(t,w)$ defines a rational curve.

(1) Find a polynomial $P(u)$ such that $t=\\frac{P(-u)}{P(u)}$.

(2) From $P(u)$, define a weighted homogeneous polynomial $f(x_1,x_2,x_3)=x_3f_1(x_1,x_2,x_3)$ of three variables $x_1,x_2,x_3$, where $(1,2,n)$ is the weight system of $(x_1,x_2,x_3)$ with $n=\\deg P(u)$. The hypersurface $D:f(x)=0$ is a free divisor in ${\\bf C}^3$. Note that $\\deg_{x_3}f_1=2$.

(3) Construct a holonomic system ${\\sl M}$ on ${\\bf C}^3$ of rank two with singularities along $D$.

(4) Construct an ordinary differential equation from the holonomic system ${\\sl M}$ with respect to $x_3$. This differential equation has three singular points $z_0,z_1,a_s$ in $x_3$-line.

(5) Putting $t=\\frac{z_1}{z_0},\\lambda=\\frac{a_s}{z_0}$, we conclude that $(t,\\lambda)$ is equivalent to the pair $(t,w)$.

Our study starts with showing the existence of $P(u)$ in (1). From the classification by Losovyy and Tykhyy, I find that the existence of $P(u)$ is guaranteed for Solutions III, IV, Solutions $k$ ($1\\le k\\le 21$, $k\\not= 4,13,14,20$) and Solution 30. We checked whether (1)-(5) are true or not in these cases separately and as a consequence (1)-(5) hold for the all these cases except Solutions 19, 21.