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Minoration de la hauteur normalisée des hypersurfaces

Francesco AmorosoSinnou David — 2000

Acta Arithmetica

1. Introduction. Dans un article célèbre, D. H. Lehmer posait la question suivante (voir [Le], §13, page 476): «The following problem arises immediately. If ε is a positive quantity, to find a polynomial of the form: f ( x ) = x r + a 1 x r - 1 + + a r where the a’s are integers, such that the absolute value of the product of those roots of f which lie outside the unit circle, lies between 1 and 1 + ε (...). Whether or not the problem has a solution for ε < 0.176 we do not know.» Cette question, toujours ouverte, est la source...

Distribution des points de petite hauteur dans les groupes multiplicatifs

Francesco AmorosoSinnou David — 2004

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We prove a new lower bound for the height of points on a subvariety  V of a multiplicative torus, which lie outside the union of torsion subvarieties of  V . Although lower bounds for the heights of these points where already known (decreasing multi-exponential function of the degree for Scmhidt and Bombieri–Zannier, [Sch], [Bo-Za], and inverse monomial in the degree by the second author of this note and P. Philippon, [Da-Phi]), our method proves the sharpest conjectures that can be formulated.

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