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

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.