The Density of Rational Points on Non-Singular Hypersurfaces, I

For any n 3, let F ZX₀, ..., X_n] be a form of degree d 5that defines a non-singular hypersurface X Pⁿ. The main resultin 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 , for any > 0. The implied constantin this estimate depends at most upon d, and n. New estimatesare also obtained for the number of representations of a positiveinteger as the sum of three dth powers, and for the paucityof integer solutions to equal sums of like polynomials. 2000Mathematics Subject Classification 11G35 (primary), 11P05, 14G05(secondary).