ZERO ORDER ESTIMATES FOR ANALYTIC FUNCTIONS

In this article we develop an important tool in transcendental number theory. More precisely, we study multiplicity estimates (or multiplicity lemmas) for analytic functions. Our main theorem reduces multiplicity estimates at zero to the study of ideals in polynomial ring stable under an appropriate map. In particular, in the case of algebraic morphisms this result gives a new link between the theory of polarized algebraic dynamical systems and transcendental number theory. Specialized to the case of differential operators this theorem improves Nesterenko's conditional result on solutions of systems of differential equations. We also deduce an analog of Nesterenko's theorem for Mahler's functions and for solutions of q-difference equations. Further, analyzing stable ideals we prove the unconditional optimal result in the case of linear functional systems of generalized Mahler's type. The latter result generalizes a famous theorem of Nishioka (1986) previously conjectured by Mahler (1969). This new multiplicity estimate allows to prove new results on algebraic independence and on measures of algebraic independence, as done in Zorin (2010 and 2011).

APPROXIMATIONS FONCTIONNELLES DES COURBES DES ESPACES PROJECTIFS

Algebraic approximation to points in projective spaces offers a new and more flexible approach to algebraic independence theory. When working over the field of algebraic numbers, it leads to open conjectures in higher dimension extending known results in Diophantine approximation. We show here that over the algebraic closure of a function field in one variable, the analog of these conjectures is true. We also derive transfer lemmas which have applications in the study of multiplicity estimates, for example.