- 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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- Publication date:
- 2005
- URN:
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uuid:3b42f877-6877-4048-bc0b-1dfadbf85b41
- Local pid:
- oai:eprints.maths.ox.ac.uk:186
- Copyright date:
- 2005
Journal article
Counting rational points on hypersurfaces
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