Small value estimates for the additive group

TL;DR

This paper extends Gel'fond's criterion to evaluate algebraic independence by analyzing polynomial derivatives on large subsets of additive and multiplicative complex subgroups, broadening its applicability.

Contribution

It introduces a generalized criterion for algebraic independence involving polynomial derivatives on large subsets of complex subgroups, including additive and multiplicative cases.

Findings

Generalization of Gel'fond's criterion to additive groups

Extension to subgroups of the multiplicative complex numbers

New estimates for polynomial derivatives on large subsets

Abstract

We generalize Gel'fond's criterion of algebraic independence to the context of a sequence of polynomials whose first derivatives take small values on large subsets of a fixed subgroup of the additive group of complex numbers, instead of just one point. We also provide one extension dealing with a subgroup of the multiplicative group of non-zero complex numbers.