Counting Rational Points on Smooth Hypersurfaces with High Degree

Abstract Let $X$ be a smooth projective hypersurface defined over $\mathbb{Q}$. We provide new bounds for rational points of bounded height on $X$. In particular, we show that if $X$ is a smooth projective hypersurface in $\mathbb{P}^{n}$ with $n\geqslant 4$ and degree $d\geqslant 50$, then the set of rational points on $X$ of height bounded by $B$ have cardinality $O_{n,d,\varepsilon }(B^{n-2+\varepsilon })$. If $X$ is smooth and has degree $d\geqslant 6$, we improve the dimension growth conjecture bound. We achieve an analogue result for affine hypersurfaces whose projective closure is smooth.