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 $n\ge 4$ and degree $d\ge 50$, then the set of rational points on $X$ of height bounded by $B$ have cardinality $O_{n,d,\epsilon }(B^{n-2+\epsilon })$. If $X$ is smooth and has degree $d\ge 6$, we improve the dimension growth conjecture bound. We achieve an analogue result for affine hypersurfaces whose projective closure is smooth.

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