Minimal algebraic solutions of the sixth Painlevé equation

R. Conte<sup>ab

a</sup> Université Paris-Saclay, ENS Paris-Saclay, CNRS, Centre Borelli, Gif-sur-Yvette, France
^b Department of Mathematics, The University of Hong Kong, Pokfulam, Hong Kong

Abstract: For each of the forty-eight exceptional algebraic solutions $u(x)$ of the sixth Painlevé equation, we build the algebraic curve $P(u,x)=0$ of a degree conjectured to be minimal, and then we give an optimal parametric representation of it. This degree is equal to the number of branches, except for fifteen solutions.

Keywords: sixth Painlevé equation, algebraic solutions.

Received: 01.02.2025
Revised: 01.02.2025

DOI: 10.4213/tmf10918

 English version: Theoretical and Mathematical Physics, 2025, 224:3, 1595–1612

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