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On the algebraic independence of generic Painlevé transcendents

Published online by Cambridge University Press:  10 March 2014

Joel Nagloo
Affiliation:
Department of Pure Mathematics, University of Leeds, UK email [email protected]
Anand Pillay
Affiliation:
Department of Mathematics, University of Notre Dame, IN 46556, USA email [email protected]
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Abstract

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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.

Type
Research Article
Copyright
© The Author(s) 2014 

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