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We show that the tritronqu\'ee solution of the Painlev\'e equation $\P1$, $y"=6y^2+z$ which is analytic for large $z$ with $ \arg z \in (-\frac{3\pi}{5},\pi)$ is pole-free in a region containing the full sector ${z \ne 0, \arg z \in[-\frac{3\pi}{5}, \pi]}$ and the disk ${z: |z| < 37/20}$. This proves inparticular the Dubrovin conjecture, an open problem in the theory of Painlev\'etranscendents. The method, building on a technique developed in Costin, Huang,Schlag (2012), is general and constructive. As a byproduct, we obtain the valueof the tritronqu\'ee and its derivative at zero within less than 1/100 rigorouserror bounds.

Proof of the Dubrovin conjecture and analysis of the tritronqu\'ee solutions of $P_I$