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On the algebraic independence of generic Painlevé transcendents

Part of: Curves Model theory Differential equations in the complex domain

Published online by Cambridge University Press:  10 March 2014

Joel Nagloo  and
Anand Pillay
Joel Nagloo
Affiliation:
Department of Pure Mathematics, University of Leeds, UK email mmjcn@leeds.ac.uk
Anand Pillay
Affiliation:
Department of Mathematics, University of Notre Dame, IN 46556, USA email apillay@nd.edu
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Abstract

We prove that if $y''=f(y,y',t,\alpha ,\beta ,\ldots)$ is a generic Painlevé equation from among the classes II, IV and V, and if $y_1,\ldots,y_n$ are distinct solutions, then $\mathrm{tr.deg}(\mathbb{C}(t)(y_1,y'_1,\ldots,y_n,y'_n)/\mathbb{C}(t))=2n$. (This was proved by Nishioka for the single equation $P_{{\rm I}}$.) For generic Painlevé III and VI, we have a slightly weaker result: $\omega $-categoricity (in the sense of model theory) of the solution space, as described below. The results confirm old beliefs about the Painlevé transcendents.

Keywords

Painlevé equationsalgebraic independencemodel theorydifferentially closed fields

MSC classification

Secondary: 14H05: Algebraic functions; function fields 14H70: Relationships with integrable systems 34M55: Painlevé and other special equations; classification, hierarchies; 03C60: Model-theoretic algebra
Type
Research Article
Information
Compositio Mathematica , Volume 150 , Issue 4 , April 2014 , pp. 668 - 678
Copyright
© The Author(s) 2014 

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