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

摘要

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 _(C d) ~(B2/d) + ~(d2), where the point is that for fixed d the constant _(C d) 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 _(O d,ε)(~(B2/d +ε)). It is known that Heath-Brown's theorem is sharp apart from the ε; our results show that for our degree d rational curves it is true with ε = 0.