scholarly journals Algebraic independence of generic Painlevé transcendents: PIII and PVI

2019 ◽  
Vol 52 (1) ◽  
pp. 100-108
Author(s):  
Joel Nagloo
Keyword(s):  
Algebraic Independence ◽  
Painlevé Transcendents ◽  
Painleve Transcendents

scholarly journals On the algebraic independence of generic Painlevé transcendents

2014 ◽  
Vol 150 (4) ◽  
pp. 668-678 ◽  
Author(s):  
Joel Nagloo ◽  
Anand Pillay
Keyword(s):  
Model Theory ◽  
Algebraic Independence ◽  
Solution Space ◽  
Single Equation ◽  
Painlevé Equation ◽  
Painlevé Transcendents ◽  
Painleve Transcendents ◽  
Alpha Beta ◽  
Painleve Equation

AbstractWe 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.

The meromorphic nature of the sixth Painlevé transcendents

2004 ◽  
Vol 94 (1) ◽  
pp. 319-342 ◽  
Author(s):  
Aimo Hinkkanen ◽  
Ilpo Laine
Keyword(s):  
Painlevé Transcendents ◽  
Painleve Transcendents

scholarly journals Lower Estimates for the Growth of Painlevé Transcendents

2003 ◽  
Vol 46 (2) ◽  
pp. 287-295 ◽  
Author(s):  
Shun SHIMOMURA
Keyword(s):  
Painlevé Transcendents ◽  
Painleve Transcendents

Painlevé transcendents

2016 ◽  
pp. 390-429
Author(s):  
Richard Beals ◽  
Roderick Wong
Keyword(s):  
Painlevé Transcendents ◽  
Painleve Transcendents

Asymptotic Behavior of the Fourth Painlevé Transcendents in the Space of Initial Values

2016 ◽  
Vol 44 (2) ◽  
pp. 195-231 ◽  
Author(s):  
Nalini Joshi ◽  
Milena Radnović
Keyword(s):  
Asymptotic Behavior ◽  
Painlevé Transcendents ◽  
Initial Values ◽  
Painleve Transcendents

Meromorphic Painlevé transcendents at a fixed singularity

2013 ◽  
Vol 286 (8-9) ◽  
pp. 861-875 ◽  
Author(s):  
Kazuo Kaneko ◽  
Yousuke Ohyama
Keyword(s):  
Painlevé Transcendents ◽  
Fixed Singularity ◽  
Painleve Transcendents

Self-duality and the Painleve transcendents

Nonlinearity ◽  
1993 ◽  
Vol 6 (4) ◽  
pp. 569-581 ◽  
Author(s):  
L J Mason ◽  
N M J Woodhouse
Keyword(s):  
Painlevé Transcendents ◽  
Self Duality ◽  
Painleve Transcendents
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