Mildness and the density of rational points on certain transcendental curves (Q628804)

This paper concerns the density of rational points on certain real-analytic non-algebraic plane curves. The curves in question are restricted Pfaffian, namely, one-dimensional subsets of the plane definable in the o-minimal structure \({\mathbf R}_{\text{Res\,Pfaff}}\). The main theorem asserts that, for a non-algebraic curve \(X\) of this type, the number \(N(X,T)\) of rational points on the curve having both coordinates of height bounded by \(H\) is bounded by \(c(\log H)^\gamma\) for suitable constants \(c,\gamma\) depending on \(X\). The proof depends on the possibility of parameterising the curve in a suitable way, and is obtained by modifying the approach of \textit{J. Pila} [Comment. Math. Univ. St. Pauli 55, No.~1, 1--8 (2006; Zbl 1129.11029)], where similar results are obtained for a more restricted type of curve definable in \({\mathbf R}_{\text{Pfaff}}\).