Some cases of Wilkie's conjecture (Q2903268)

Wilkie has conjectured a very strong upper bound for the number of the rational points up to a given height on a set definable in the o-minimal structure \({\mathbb R}_{\exp}\) that do not lie on some positive-dimensional semialgebraic subset. Specifically, a bound polynomial in the logarithm of the height. This paper affirms the conjecture for curves (i.e., definable sets of dimension 1) in this structure, and for certain surfaces. For the case of surfaces, the main issue is to establish that they can be parameterised in a suitably controlled way. The case of curves was obtained independently by \textit{G. O. Jones} and \textit{M. E. M. Thomas} [Q. J. Math. 63, No. 3, 637--651 (2012; Zbl 1253.03065)], who also affirm the conjecture for some different surfaces.