Journal article

Counting rational points on hypersurfaces

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 $\varepsilon$.

Access Document

Files:
• (pdf, 298.1kb)

Authors

D. R. Heath-Brown More by this author
T. D. Browning More by this author
Publication date:
2005
URN: