The density of rational points on non-singular hypersurfaces, II - Heath-Brown, D; Browning, T (2006)
For any integers $d,n ≥ 2$, let $X \subset \mathbb{P}^{n}$ be a non-singular hypersurface of degree $d$ that is defined over $\mathbb{Q}$. The main result in this paper is a proof that the number $N_X(B)$ of $\mathbb{Q}$-rational points on $X$ which have height at most $B$ satisfies $ N_X(B)=O_{d,ε,n}(B^{n-1+ε}), $ for any $ε>0$. The implied constant in this estimate depends at most upon $d, ε$ and $n$.