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

Browning, T. D.; Heath-Brown, D. R.

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

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Description

Let $X \subset \mathbb{P}^n$ be a non-singular hypersurface of degree $d>1$, and let $ε>0$. This paper is concerned with the conjecture that there are $O(B^{n-1+ε})$ rational points on $X$ that have height at most $B$, in which the implied constant is allowed to depend only upon $d, ε$ and $n$. In particular this conjecture is shown to hold as soon as $d>4$. Furthermore, the main ideas in the proof are used to obtain new paucity estimates for certain diophantine equations.
12 pages