On the irreducibility of the first differential equation of Painlevé. (English) Zbl 0704.12007

Umemura, Hiroshi

Algebraic geometry and commutative algebra, in Honor of Masayoshi Nagata, Vol. II, 771-789 (1988).

[For the entire collection see Zbl 0655.00011.]
This paper contains an almost self-contained proof of the irreducibility of the differential equation \(y''=6x^ 2+x\), in a sense specified in the paper and coinciding precisely with Painlevé’s expectations. The only reference actually used is some previous work of the author, recalled in the first section, dealing with two equivalent methods of generating differential fields. No use is made of Kolchin’s differential Galois theory. Instead, an interesting interpretation of the invariant vector field tangent at some point to a curve on an algebraic group is given, and related to the differential properties of the residue field at the generic point of such a curve.
The results of the paper do not appear to be totally original, but the geometric arguments used in the proofs make the exposition interesting and pleasant.

Reviewer: F.Baldassarri

MSC:

12H05	Differential algebra
34M55	Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies

Keywords:

irreducibility; differential fields; algebraic group

Citations:

Zbl 0655.00011