For any integers d,n ≥ 2, let X ⊂ Pⁿ be a non-singular hypersurface of degree d that is defined over the rational numbers. The main result in this paper is a proof that the number of rational points on X which have height at most B is O(Bn - 1 + ε), for any ε > 0. The implied constant in this estimate depends at most upon d, ε and n.
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Browning et al. (2006) studied this question.