Zero Order Estimates for Analytic Functions

TL;DR

This paper introduces a general multiplicity estimate linking algebraic dynamical systems and transcendental number theory, improving existing results and providing universal tools for algebraic independence across various fields.

Contribution

It develops a new multiplicity estimate reducing proofs to stable ideals, connecting dynamical systems with transcendental number theory, and generalizes key theorems in functional and differential systems.

Findings

Established a new multiplicity estimate applicable to various fields.

Generalized Nishioka's theorem for linear functional systems.

Improved Nesterenko's results on solutions of differential equations.

Abstract

The primary goal of this paper is to provide a general multiplicity estimate. Our main theorem allows to reduce a proof of multiplicity lemma to the study of ideals stable under some appropriate transformation of a polynomial ring. In particular, this result leads to a new link between the theory of polarized algebraic dynamical systems and transcendental number theory. On the other hand, it allows to establish an improvement of Nesterenko's conditional result on solutions of systems of differential equations. We also deduce, under some condition on stable varieties, the optimal multiplicity estimate in the case of generalized Mahler's functional equations, previously studied by Mahler, Nishioka, Topfer and others. 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), and simultaneously it gives a counterpart in the case of functional systems for an important unconditional result of Nesterenko (1977) concerning linear differential systems. In summary, we provide a new universal tool for transcendental number theory, applicable with fields of any characteristic. It opens the way to new results on algebraic independence, as shown in Zorin (2010).