Counting rational points on smooth cyclic covers

A conjecture of Serre concerns the number of rational points of bounded height on a finite cover of projective space ${\mathbb{P}}^{n?1}$. In this paper, we achieve Serre?s conjecture in the special case of smooth cyclic covers of any degree when $n?10$, and surpass it for covers of degree $r?3$ when $n>10$. This is achieved by a new bound for the number of perfect r-th power values of a polynomial with nonsingular leading form, obtained via a combination of an r-th power sieve and the q-analogue of van der Corput?s method.