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Tanveer","submitted_at":"2012-09-05T15:11:24Z","abstract_excerpt":"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. 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