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 Pn−1. In this paper, we achieve Serreʼs 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 Corputʼs method.