ON UNIFORM BOUNDS FOR RATIONAL POINTS ON RATIONAL CURVES AND THIN SETS

We show that, for any epsilon > 0, the number of rational points of height less than B on the image of a quadratic map from P-1 to P-n (n >= 1), under certain conditions, is bounded above by CB/| R |(1/(4n(n+1))) +C-epsilon| R |(epsilon), where the point is that the constant C is independent of the choice of the map. R is the resultant of the map and is a nonzero integer. In the special case of quadratic plane curves, we prove a bound of CB/| R |(1/72) + 4 which improves on a result of Browning and Heath-Brown by establishing an inverse dependence on the resultant. Heath-Brown proved that for any epsilon > 0, the number of rational points of height less than B on a degree d plane curve is O-epsilon,O-d(B2/d+epsilon). Browning and Heath-Brown later proved that this result holds with epsilon = 0 for quadratic curves. It is known that Heath-Brown's theorem is sharp apart from the epsilon, and in fact, Ellenberg and Venkatesh have proved that there is some delta > 0 (depending only on d) such that the point counting function for any plane curve of positive genus is O-d(B2/d-delta) Our results shed light on the open question of whether Heath-Brown's theorem is true with epsilon = 0.