The density of rational points on non-singular hypersurfaces, I

For any n >= 3, let F is an element of Z[X-0,..., X-n] be a form of degree d >= 5 that defines a non-singular hypersurface X subset of P-n. The main result in this paper is a proof of the fact that the number N(F; B) of Q-rational points on X which have height at most B satisfies N(F; B) = O-d,O-epsilon,O-n (Bn-1+epsilon) for any epsilon > 0. The implied constant in this estimate depends at most upon d, epsilon and n. New estimates are also obtained for the number of representations of a positive integer as the sum of three dth powers, and for the paucity of integer solutions to equal sums of like polynomials.