Skip to main content Accessibility help
×
Home

On the algebraic independence of generic Painlevé transcendents

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

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.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      On the algebraic independence of generic Painlevé transcendents
      Available formats
      ×

      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      On the algebraic independence of generic Painlevé transcendents
      Available formats
      ×

      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      On the algebraic independence of generic Painlevé transcendents
      Available formats
      ×

Copyright

References

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.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

Keywords

MSC classification

Related content

Powered by UNSILO

On the algebraic independence of generic Painlevé transcendents

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

Metrics

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.