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QUANTITATIVE PROPERTIES OF MEROMORPHIC SOLUTIONS TO SOME DIFFERENTIAL-DIFFERENCE EQUATIONS
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 99 / Issue 2 / April 2019
- Published online by Cambridge University Press:
- 26 December 2018, pp. 250-261
- Print publication:
- April 2019
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On the algebraic independence of generic Painlevé transcendents
- Part of
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- Journal:
- Compositio Mathematica / Volume 150 / Issue 4 / April 2014
- Published online by Cambridge University Press:
- 10 March 2014, pp. 668-678
- Print publication:
- April 2014
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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.
