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The $abc$ conjecture predicts a highly non trivial upper bound for the heightof an algebraic point in terms of its discriminant and its intersection with afixed divisor of the projective line counted without multiplicity. We describethe two independent proofs of the strong $abc$ conjecture over function fieldsgiven by McQuillan and Yamanoi. The first proof relies on tools fromdifferential and algebraic geometry; the second relies on analytic andtopological methods. They correspond respectively to the Nevanlinna and theAhlfors approach to the Nevanlinna Second Main Theorem.

The strong $ABC$ conjecture over function fields (after McQuillan and Yamanoi)