Some remarks about algebraic independence measures in high dimension
Polynomials with high multiplicity
Bounds for the degrees in the membership test for a polynomial ideal
Values of polynomials with integer coefficients and distance to their common zeros
Multiplicité et formes éliminantes
Small points on a multiplicative group and class number problem
Let be an algebraic subvariety of a torus and denote by the complement in of the Zariski closure of the set of torsion points of . By a theorem of Zhang, is discrete for the metric induced by the normalized height . We describe some quantitative versions of this result, close to the conjectural bounds, and we discuss some applications to study of the class group of some number fields.
Une minoration pour l'exposant du groupe de classes d'idéaux
- transfinite diameter and number theoretic applications
Sur le diamètre transfini entier d'un intervalle réel
En utilisant à la fois la théorie des polynômes orthogonaux et des arguments élémentaires de géométrie des nombres, nous donnons ici des nouveaux encadrements pour le diamètre transfini entier d’un intervalle d’extrémités rationnelles. Ces encadrements dépendent explicitement de la longueur de et des dénominateurs de ses extrémités.
Minoration de la hauteur normalisée des hypersurfaces
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: 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...
Upper bounds for the coefficients of irreducible integer polynomials in several variables
On the zeros of linear recurrence sequences
On the distribution on the roots of polynomials
Using classical results on conjugate functions, we give very short proofs of theorems of Erdös–Turán and Blatt concerning the angular distribution of the roots of polynomials. Then we study some examples.
A relative Dobrowolski lower bound over abelian extensions
Approximation measures for logarithms of algebraic numbers
Distribution des points de petite hauteur dans les groupes multiplicatifs
We prove a new lower bound for the height of points on a subvariety of a multiplicative torus, which lie outside the union of torsion subvarieties of . 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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