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Birational automorphism groups and differential equations. (English) Zbl 0714.12009

The classical theory of algebraic differential equations has been significantly inspired by the deep work of P. Painlevé about the beginning of this century. In his celebrated “Stockholm Lectures” [cf. P. Painlevé, Leçons de Stockholm, in: Oevres de Paul Painlevé, Tome I, Editions du Centre National de la Recherche Scientifique (Paris 1973), p. 199-818], Painlevé elaborated the link between algebraic differential equations, differential algebraic function fields, the algebraic varieties associated with them, and the structure of the birational automorphism groups of those varieties. Applying this interplay to integrating differential equations, he showed that all algebraic differential equations of the form \(y''=R(y',y,x)\) are solvable by the so far known functions, or those which are constructable from them by rational processes (including differentiation and integration), apart from six exceptions - the so-called Painlevé equations. The simplest among them is the equation \(y''=6y^ 2+x\), but its irreducibility (in the above sense), claimed by Painlevé, has - up to very recently [cf. the author, Nagoya Math. J. 117, 125-171 (1990; Zbl 0688.34006)] - never been proved rigorously.
The present paper is the (delayed) published version of the author’s attempt, undertaken in 1984/85, to clarify Painlevé’s work systematically and rigorously, just to find an access to prove the irreducibility of Painlevé’s first equation thoroughly. This has been done, in the meantime (cf. the paper cited above), and the recent progress in studying Painlevé’s first equation is, in fact, based upon the systematic account on the general Painlevé theory given in the present, already well-known treatise. Actually, this paper may be regarded as a fundamental source of the following recent, furthergoing articles dealing with Painlevé equations:
(1) H. Umemura, On the irreducibility of the first differential equation of Painlevé, in: Algebraic geometry and commutative algebra, Vol. II, 771-789 (1988; Zbl 0704.12007); (2) K. Nishioka, A note on the transcendency of Painlevé’s first transcendent, Nagoya Math. J. 109, 63-67 (1988; Zbl 0613.34030); (3) K. Nishioka, General solutions depending algebraically on arbitrary constants, ibid. 113, 1-6 (1989; Zbl 0702.12008); (4) K. Nishioka, Differential algebraic function fields depending rationally on arbitrary constants, ibid. 113, 173-179 (1989; Zbl 0695.12016); (5) H. Umemura, Second proof of the irreducibility of the first differential equation of Painlevé, ibid. 117, 125-171 (1990; Zbl 0688.34006).
As for the content of the present, finally published version of the contemporary interpretation of Painlevé’s approach to algebraic differential equations, it consists of a thorough and consequent algebro- geometric foundation of Painlevé’s ideas, concepts, and methods developed in his Stockholm lectures. The whole presentation is based upon the modern framework of algebraic and complex-analytic geometry, in the spirit of Grothendieck’s E.G.A. and Serre’s G.A.G.A., and as such highly self-contained, i.e., also accessible for non-algebraists. Part I of the paper is devoted to Painlevé’s theorem on the relation between analytic subgroups and algebraic subgroups, respectively, of the birational automorphism group of a complex algebraic variety, whereas Part II provides a comprehensive and rigorous treatment of systems of Pfaffian differential equations over complex manifolds, their differential- algebraic aspects, and Painlevé’s solvability theorems for algebraic differential equations. Altogether, the present work is certainly of fundamental importance in the field of algebraic analysis.
Reviewer: W.Kleinert

MSC:

12H20 Abstract differential equations
14E07 Birational automorphisms, Cremona group and generalizations
12-03 History of field theory
34M55 Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies
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References:

[1] Leçons de Stockholm, OEuvres de P. Painlevé I pp 199– (1972)
[2] Grund. der math. Wiss 221 (1976)
[3] Math. Ann. 209 (1974)
[4] les schemas de Hilbert, Sém. Bourbaki, 1.13, 1960/61
[5] Rapport sur les théorèmes de finitude de Grauert et Remmert Sém. H. Cartan 13e année 1960/61
[6] Nagoya Math. J. 79 pp 47– (1980) · Zbl 0412.14007
[7] Ann. Sci. Ecole Norm. Sup., 4e série, t 3 (1970)
[8] Linear algebraic groups (1969)
[9] Publ. of Math. Soc. of Japan (1958)
[10] DOI: 10.2307/2372535 · Zbl 0065.14201
[11] Algebraic and Topological Theories–to the memory of Dr. Takehiko MIYATA pp 467– (1985)
[12] Calcul infinitesimal, Collection Methodes (1968)
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