Proof of the Dubrovin conjecture and analysis of the tritronquée solutions of PI

Abstract

We show that the tritronquée solution $y_t$ of the Painlevé equation $P_I$ that behaves algebraically for large $z$ with $argz=\pi /5$ is analytic in a region containing the sector $\{z\ne 0,argz\in [-3\pi /5,\pi \left]\right\}$ and the disk $\{z:|z|<37/20\}$. This implies the Dubrovin conjecture, an important open problem in the theory of Painlevé transcendents. As a by-product, we obtain the value of the tritronquée and its derivative at zero, also important in applications, within less than $1/100$ rigorous error bounds.