On uniform bounds for rational points on rational curves of arbitrary degree

We show that for any ϵ>0 the number of rational points of height less than B on the image of a degree d map from P1 to P2 is bounded above by CdB2/d+d2, where the point is that for fixed d the constant Cd is independent of the choice of map. This improves on a result of Heath-Brown, which states that for any ϵ>0 the number of rational points of height less than B on a degree d plane curve is Od,ϵ(B2/d+ϵ). It is known that Heath-Brownʼs theorem is sharp apart from the ϵ; our results show that for our degree d rational curves it is true with ϵ=0.