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Let A, B, C = A + B be relatively prime polynomials with complex coefficients and maximal degree D (&#x3E;0). The Mason’s theorem implies that D + 1 does not exceed the number r (ABC) of distinct roots of the product ABC. Similarly, let A, B, C = A + B be relatively prime positive integers and S = max(A, B, C). Let r(ABC) be the product of all primes dividing the product ABC. The abc-conjecture implies that, for any 03B5 &#x3E; 0, there exists C03B5 &#x3E; 0 such that the inequality: r(ABC) &#x3E; C03B5S1-03B5 holds for any triple A, B, C = A + B of integers. The cases of equality r(ABC) = D + 1 for polynomials A, B, C = A + B are linked to numerous results in number theory ; triples of integers generated by these cases lead, by using the abc-conjecture, to optimal minoration of r(G(A,B)) (where G ~ Z [X, T] is a form and A, B are coprime integers) ; in these polynomial constructions of integers, the role of the Mason’s theorem is crucial.

Imbrications entre le théorème de Mason, la descente de Belyi et les différentes formes de la conjecture $(abc)$