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

Let $X \subset \mathbb{P}^n$ be a non-singular hypersurface of degree $d>1$, and let $\epsilon>0$. This paper is concerned with the conjecture that there are $O(B^{n-1+\epsilon})$ rational points on $X$ that have height at most $B$, in which the implied constant is allowed to depend only upon $d, \epsilon$ and $n$. In particular this conjecture is shown to hold as soon as $d>4$. Furthermore, the main ideas in the proof are used to obtain new paucity estimates for certain diophantine equations.