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

Abstract

We show that if a sequence of non-zero polynomials in [Formula: see text] takes small values at translates of a fixed point [Formula: see text] by multiples of a fixed rational point within the group [Formula: see text], then [Formula: see text] and [Formula: see text] are both algebraic over [Formula: see text]. 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.