Minimal algebraic solutions of the sixth equation of Painlevé

Abstract

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

19 pages, no figure, to appear, Theoretical and mathematicalphysics