Multiplicity Estimates for Algebraically Dependent Analytic Functions

TL;DR

This paper introduces a new general multiplicity estimate applicable to sets of functions regardless of their algebraic independence, enhancing tools for measuring algebraic independence of functions and their values.

Contribution

It provides a novel multiplicity estimate that does not assume algebraic independence, aiding proofs of algebraic independence of complex numbers and functions.

Findings

New multiplicity estimate applicable without independence assumptions

Improves tools for algebraic independence proofs

Provides measures of algebraic independence over finite fields

Abstract

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 Mahler's 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 ${\bf F}_q[[T]]$, where ${\bf F}_q$ denotes a finite field.