Counting rational points on algebraic varieties

Abstract

For any $N\ge 2$, let $Z\subset {\mathbb{P}}^N$ be a geometrically integral algebraic variety of degree $d$. This article is concerned with the number $N_Z(B)$ of $\mathbb{Q}$-rational points on $Z$ which have height at most $B$. For any $\epsilon >0$, we establish the estimate ${\displaystyle N_Z(B)=O_{d,\epsilon ,N}(B^{\dim Z+\epsilon })}$, provided that $d,\epsilon$. As indicated, the implied constant depends at most on $d,\epsilon$, and $N$