Minimal algebraic solutions of the sixth equation of Painlevé

TL;DR

This work addresses the problem of representing each of the 48 exceptional algebraic ${\rm P_{VI}}$ solutions by a minimal algebraic curve $P(u,x)=0$ whose global degree $d$ matches the number of branches $b$. By exploiting the symmetry group of sign changes, projective homographies, and a unique birational folding, the authors reduce $d$ to its minimal value, ideally $d=b$, for representative curves across genus 0, genus 1, and higher-genus folded cases. They provide explicit, compact parametric representations: genus-zero cases yield simple rational parametrizations; genus-one cases adopt an elliptic representation with $x=\tfrac{1}{2}+R_1(s)t$ and $u=\tfrac{1}{2}+R_2(s)t$ on a Weierstrass form, while higher-genus folded solutions involve hyperelliptic or multi-curve representations with fixed-pole and symmetry considerations. The results catalog which classes admit $d=b$, present detailed examples (e.g., I21, I27, I38, I50–I52), and deliver minimal, structurally uniform forms that facilitate computation and applications in mathematical physics and the theory of Painlevé equations.

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.