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

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Michel Langevin
Imbrications entre le théorème de Mason, la descente de Belyi et les différentes formes de la conjecture $(abc)$
Journal de théorie des nombres de Bordeaux, 11 no. 1 (1999), p. 91-109, doi: 10.5802/jtnb.240
Article PDF | Reviews MR 1730434 | Zbl 0983.11015 | 1 citation in Cedram

Résumé - Abstract

Let $A, B, C = A + B$ be relatively prime polynomials with complex coefficients and maximal degree $D$ (> $0$). 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 $\epsilon > 0$, there exists $C_ \epsilon > 0$ such that the inequality: $r(ABC) \ge C_\epsilon S^{1-\epsilon }$ 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 $G \in \mathbb{Z} [X, T]$ is a form and $A, B$ are coprime integers) ; in these polynomial constructions of integers, the role of the Mason’s theorem is crucial.

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