A small value estimate for Ga x Gm

TL;DR

This paper establishes a small value estimate for the product of an additive and a multiplicative algebraic group, providing conditions under which certain points are algebraic, with implications for transcendental number theory.

Contribution

It presents a new small value estimate for Ga x Gm, improving upon existing criteria for algebraic independence and aiding transcendental number theory applications.

Findings

If polynomials take small values and derivatives at a point, then the point is algebraic.

The estimate involves growth conditions on polynomials' degree, norm, and derivatives.

Results compare favorably with Dirichlet's principle and Philippon's criterion.

Abstract

A small value estimate is a statement providing necessary conditions for the existence of certain sequences of non-zero polynomials with integer coefficients taking small values at points of an algebraic group. Such statements are desirable for applications to transcendental number theory to analyze the outcome of the construction of an auxiliary function. In this paper, we present a result of this type for the product Ga x Gm whose underlying group of complex points is C x C*. It shows that if a certain sequence of non-zero polynomials in Z[X,Y] take small values at a point (xi,eta) together with their first derivatives with respect to the invariant derivation of the group, then both xi and eta are algebraic over Q. The precise statement involves growth conditions on the degree and norm of these polynomials as well as on the absolute values of their derivatives. It improves on a direct application of Philippon's criterion for algebraic independence and compares favorably with constructions coming from Dirichlet's box principle.