Abstract:

For any $n \geq 2$, let $F \in \mathbb{Z}[x_1,\ldots,x_n]$ be a form of degree $d\geq 2$, which produces a geometrically irreducible hypersurface in $\mathbb{P}^{n-1}$. This paper is concerned with the number $N(F;B)$ of rational points on $F=0$ which have height at most $B$. For any $\varepsilon>0$ we establish the estimate $ N(F;B)=O(B^{n-2+\varepsilon}),$ whenever either $n\leq 5$ or the hypersurface is not a union of lines. Here the implied constant depends at most upon $d, n$ and $\varepsilon$.

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