Simultaneous Diophantine approximation to points on the Veronese curve

We compute the Hausdorff dimension of the set of simultaneously q-λ-well approximable points on the Veronese curve for λ between 1/n and 2/2n-1. For λ≤ 2/n we provide an upper bound for this Hausdorff dimension. In the course of the proof we establish that the number of cubic polynomials of height at most H and non-zero discriminant at most D is bounded from above by c(ε) H2/3 + ε D5/6.