Skip to main content
Log in

Picard and Chazy solutions to the Painlevé VI equation

  • Original article
  • Published:
Mathematische Annalen Aims and scope Submit manuscript

Abstract.

We study the solutions of a particular family of Painlevé VI equations with parameters \(\beta=\gamma=0, \delta=\frac{1}{2}\) and \(2\alpha=(2\mu-1)^2\), for \(2\mu\in{\mathbb Z}\). We show that in the case of half-integer \(\mu\), all solutions can be written in terms of known functions and they are of two types: a two-parameter family of solutions found by Picard and a new one-parameter family of classical solutions which we call Chazy solutions. We give explicit formulae for them and completely determine their asymptotic behaviour near the singular points \(0,1,\infty\) and their nonlinear monodromy. We study the structure of analytic continuation of the solutions to the PVI\(_\mu\) equation for any \(\mu\) such that \(2\mu\in{\mathbb Z}\). As an application, we classify all the algebraic solutions. For \(\mu\) half-integer, we show that they are in one to one correspondence with regular polygons or star-polygons in the plane. For \(\mu\) integer, we show that all algebraic solutions belong to a one-parameter family of rational solutions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Author information

Authors and Affiliations

Authors

Additional information

Received: 23 February 1999 / Accepted: 10 January 2001 / Published online: 18 June 2001

Rights and permissions

Reprints and permissions

About this article

Cite this article

Mazzocco, M. Picard and Chazy solutions to the Painlevé VI equation. Math Ann 321, 157–195 (2001). https://doi.org/10.1007/PL00004500

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/PL00004500

Keywords

Navigation