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 in particular the Dubrovin conjecture, an open problem in the theory of Painlev\'e transcendents. The method, building on a technique developed in Costin, Huang, Schlag (2012), is general and constructive. As a byproduct, we obtain the value of the tritronqu\'ee and its derivative at zero within less than 1/100 rigorous error bounds.
Categories: math.CA math-ph math.MP