Evgeniy Zorin

Multiplicity estimates for algebraically dependent analytic functions

We prove a new general multiplicity estimate applicable to sets of functions without any assumption on algebraic independence. The multiplicity estimates are commonly used in determining measures of algebraic independence of values of functions, for instance within the context of Mahlers method. For this reason, our result provides an important tool for the proofs of algebraic independence of complex numbers. At the same time, these estimates can be considered as a measure of algebraic independence of functions themselves. Hence our result provides, under some conditions, the measure of algebraic independence of elements in

ZERO ORDER ESTIMATES FOR ANALYTIC FUNCTIONS

{\bf F}_q[[T]]

New results on algebraic independence with Mahlerʼs method

, where

Multiplicity estimate for solutions of extended Ramanujan’s system

{\bf F}_q

Explicit bounds for rational points near planar curves and metric Diophantine approximation

denotes a finite field.

Diophantine Approximation on Manifolds and the Distribution of Rational Points: Contributions to the Convergence Theory

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 Nesterenkos conditional result on solutions of systems of differential equations. We also deduce an analog of Nesterenkos theorem for Mahlers 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 Mahlers 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).

Diophantine approximation on curves and the distribution of rational points: divergence theory.

We give some new results on algebraic independence within Mahler’s method, including algebraic independence of values at transcendental points. We also give some new measures of algebraic independence for infinite series of numbers. In particular, our results furnishes, for n ≥ 1 arbitrarily large, new examples of sets (θ1, . . . , θn) ∈ R n normal in the sense of definition formulated by Grigory Chudnovsky (1980). Mathematics Subject Classification 2000: 11J81, 11J82, 11J61 Let p(z) = p1(z)/p2(z) be a rational fraction with coefficients in Q. We denote d = deg p, δ = ordz=0p. Let f1(z),...,fn(z) ∈ Q[[z]] be functions analytic in some neighborhood U of 0, with algebraic coefficients and satisfying the following system of functional equations a(z)f(z) = A(z)f (p(z)) +B(z), (1) where f(z) = (f1(z), . . . , fn(z)), a(z) ∈ Q[z] and A (resp. B) is a matrix n× n (resp. n× 1) with coefficients in Q[z]. Algebraic independence of values of such functions was studied by Becker, Mahler, Nishioka, Töpfer and others [1, 5, 6, 7, 10]. For this purpose one can also use a general method developed in [9] (see also [8]). This method requires multiplicity estimate. Recently a new multiplicity lemma for solutions of (1) was established (see [11], Theorem 3.11, also [12]). Using this latter result with the general method from [9] one can deduce the following theorems, which improve previously known and establish new results on algebraic independence and measures of algebraic independence. Theorem 1 Let f1(z),. . . ,fn(z) be analytic functions as described above, algebraically independent over C(z), and with additional condition p(z) ∈ Q[z]. Let y ∈ Q ∗ be such that p(y) → 0 Institut de mathématiques de Jussieu, Université Paris 7, Paris, France. E-mail: evgeniyzorin@yandex.ru

On the minimum of a positive definite quadratic form over non-zero lattice points. Theory and applications

We establish a new multiplicity lemma for solutions of a differential system extending Ramanujans classical differential relations. This result can be useful in the study of arithmetic properties of values of Riemann zeta function at odd positive integers (Nesterenko, 2011).