A small value estimate in dimension two involving translations by rational points

TL;DR

This paper proves that if certain polynomials take small values at translated points involving rational points in a complex group, then the points are algebraic, with results depending on polynomial growth conditions.

Contribution

It establishes a new small value estimate in dimension two involving translations by rational points, linking polynomial behavior to algebraicity of points.

Findings

Points are algebraic if polynomials take small values at their translates.

Results depend on growth conditions of polynomials' degree and norm.

The estimate is essentially optimal within certain parameter ranges.

Abstract

We show that, if a sequence of non-zero polynomials in $\mathbb{Z}[X_1,X_2]$ take small values at translates of a fixed point $(\xi,\eta)$ by multiples of a fixed rational point within the group $\mathbb{C}\times\mathbb{C}^*$, then $\xi$ and $\eta$ are both algebraic over $\mathbb{Q}$. The precise statement involves growth conditions on the degree and norm of these polynomials as well as on their absolute values at these translates. It is essentially best possible in some range of the parameters.