THE DENSITY OF RATIONAL POINTS ON NON-SINGULAR HYPERSURFACES, I

Abstract

For any $n \geq 3$, let $F \in \mathbb{Z}[X_0,\ldots,X_n]$ be a form of degree $d\geq 5$ that defines a non-singular hypersurface $X \subset \mathbb{P}^{n}$. The main result in this paper is a proof of the fact that the number $N(F;B)$ of $\mathbb{Q}$-rational points on $X$ which have height at most $B$ satisfies $N(F;B)=O_{d,\varepsilon,n}(B^{n-1+\varepsilon}),$ 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.