On the Irreducibility of the First Differential Equation of Painlevé

Abstract

This chapter presents a modified version of the proof due to Nishioka. A non-zero K-linear combination of monomials of weight w is called a weighted homogeneous polynomial of weight w. The weighted homogeneous polynomial of a given weight w form, together with zero, a finite dimensional K-vector space Rw. The chapter discusses the the irreducibility of the first differential equation of Painlevé. The irreducibility of the first differential equation of Painlevé is a historical problem and there are serious discussions on the legitimity of the so far given proofs and the tools used in the proof.