在這篇論文中，我們對第二 Painleve方程 (P_2)

y^' = 2 y^3 + z y + α.

的有理解做了詳細的研究。我們跟隨著 Murata 的方法去證明 P_2 有唯一的有理解若且唯若 α= n 屬於整數。接著，我們證明有理解 y_n 可以被表示為一組多項式的數列形式 (叫做 Yablonskii-Vorob'ev 多項式)，它滿足一個非線性的三項遞迴關係。亦即， y_n = d/dz log Q_{n-1}/Q_{n}，而 Q_{n} 滿足
Q_{n-1} Q_{n+1}= 4[(Q_{n}^{'})^2-Q_n Q_n^{'}] +zQ^2_{n} 伴隨著 Q_0 (z)= 1 和 Q_1 (z) =z 。 一直以來，要證明 Q_{n} 是一個多項式需要對與 P_2 相關的 τ 函數進行分析。在 2001 年，Taneda 和 Fukutani-Okamoto-Umemura 各自給出代數的證明，說明上述的遞迴關係必然生成出多項式，連帶也給出了這些多項式的性質。我們將展示他們的代數證明方法。此外，我們將把 Taneda 的方法推廣到 P_3 的 Umemura 多項式上。

In this thesis, we make a detailed study of the rational solutions of the second Painleve equations (P_2)

y^' = 2 y^3 + z y + α.

We follow Murata's method to show that P_2 has a unique rational solution if and only if α= n ∈ Z . Then we show that the rational solution y_n can be expressed in terms of a sequence of polynomials (called Yablonskii-Vorob'ev polynomials) which satisfies an nonlinear three-term recurrence relation. Namely
y_n = d/dz log Q_{n-1}/Q_{n} where
Q_{n-1} Q_{n+1}= 4[(Q_{n}^{'})^2-Q_n Q_n^{'}] +zQ^2_{n} with Q_0 (z)= 1 and Q_1 (z) =z . Traditionally, the proof that Q_{n} is a polynomial makes use of the analytic proof of a tau function related to P_2. In 2001, Taneda and Fukutani-Okamoto-Umemura independently gave algebraic proofs that the above recurrence relation would generate polynomials only, as well as properties of these polynomials. We shall give an exposition of their methods. Furthermore, we shall extend Taneda's method to work on those Umemura polynomials of P_3 .

1 Introduction 1

2 Rational solutions of second Painlev´e equation 5
2.1 The condition of rational solutions . . . . . . . . . . . . . . . . . . . . 5
2.2 B¨acklund transformation . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Yablonskii-Vorob’ev polynomials 14
3.1 Equivalent equations of P2(α) . . . . . . . . . . . . . . . . . . . . . . . 14
3.2 Rational solutions of P2(α) and YV polynomials . . . . . . . . . . . . . 15
3.3 Proof of Theorem 3.2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.4 Properties of the YV polynomials . . . . . . . . . . . . . . . . . . . . . 20

4 Algebraic methods for the special polynomials 26
4.1 Taneda’s method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.2 Fukutani-Okamoto-Umemura’s method . . . . . . . . . . . . . . . . . . 31
4.3 The Umemura polynomials for P3 from an algebraic viewpoint . . . . . 37

Appendix A A list of Yablonskii-Vorob’ev polynomials for P2 44
Appendix B A list of Umemura polynomials for P3 46

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