Skip to main content Accessibility help

On the algebraic independence of generic Painlevé transcendents

  • Joel Nagloo (a1) and Anand Pillay (a2)


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.



Hide All
[Boa06]Boalch, P., The fifty-two icosahedral solutions to Painlevé VI, J. Reine Angew. Math. 596 (2006), 183214.
[GLS02]Gromak, V., Laine, I. and Shimomura, S., Painlevé differential equations in the complex plane, De Gruyter Studies in Mathematics, vol. 28 (De Gruyter, 2002).
[KLM94]Kitaev, A., Law, C. and McLeod, J., Rational solutions of the fifth Painlevé equation, Differ. Integral Equations 7 (1994), 9671000.
[LT08]Lisovyy, O. and Tykhyy, Yu., Algebraic solutions of the sixth Painlevé equation, Preprint (2008), arXiv:0809.4873.
[Mar05]Marker, D., Model theory of differential fields, in Model theory of fields, Lecture Notes in Logic, vol. 5, eds Marker, D., Messmer, M. and Pillay, A. (Springer, Berlin–Tokyo, 2005).
[Mur85]Murata, Y., Rational solutions of the second and fourth Painlevé equations, Funkcial. Ekvac. 28 (1985), 132.
[Mur95]Murata, Y., Classical solutions of the third Painlevé equation, Nagoya Math. 139 (1995), 3765.
[NP11]Nagloo, R. and Pillay, A., On Algebraic relations between solutions of a generic Painlevé equation, Preprint (2011), arXiv:1112.2916.
[Nis04]Nishioka, K., Algebraic independence of Painlevé first transcendents, Funkcial. Ekvac. 47 (2004), 351360.
[Oka87]Okamoto, K., Studies on the Painlevé equations IV, third Painlevé equation, P III, Funkcial. Ekvac. 30 (1987), 305332.
[UW97]Umemura, H. and Watanabe, H., Solutions of the second and fourth Painlevé equations, I, Nagoya Math. J 148 (1997), 151198.
[Wat98]Watanabe, H., Birational canonical transformations and classical solutions of the sixth Painlevé equation, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) XXVII (1998), 379425.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Compositio Mathematica
  • ISSN: 0010-437X
  • EISSN: 1570-5846
  • URL: /core/journals/compositio-mathematica
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


MSC classification


Altmetric attention score

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed