We show that the tritronquée solution of the Painlevé equation that behaves algebraically for large with is analytic in a region containing the sector and the disk . 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 rigorous error bounds.
O. Costin. M. Huang. S. Tanveer. "Proof of the Dubrovin conjecture and analysis of the tritronquée solutions of ." Duke Math. J. 163 (4) 665 - 704, 15 March 2014. https://doi.org/10.1215/00127094-2429589