The density of rational points on non-singular hypersurfaces, I

Journal article

For any $n \geq 3$, let $F ∈ ℤ[X 0,⋯,Xn]$ be a form of degree $d\geq 5$ that defines a non-singular hypersurface $X ⊂ ℙ n. The main result in this paper is a proof of the fact that the number $N(F;B) of ℚ-rational points on $X$ which have height at most $B$ satisfies $N(F;B)=Od, ε,n(Bn-1+ε) for any $\varepsilon >0$. The implied constant in this estimate depends at most upon $d$, $\varepsilon$ and $n$. New estimates are also obtained for the number of representations of a positive integer as the sum of three $d$th powers, and for the paucity of integer solutions to equal sums of like polynomials. © 2006 London Mathematical Society.

Authors: Browning, T; Heath-Brown, D

• Publication status: Published
• Journal: BULLETIN OF THE LONDON MATHEMATICAL SOCIETY
• Volume: 38
• Issue: 3
• Pages: 401-410
• Publication date: 2006-06-01
• DOI: 10.1112/S0024609305018412
• EISSN: 1469-2120
• ISSN: 0024-6093
• Language: English