Small solutions of polynomial congruences

Abstract

Let p be prime and q|p − 1. Suppose x q ≡ a(mod p) has a solution. We estimate the size of the smallest solution x 0 with 0 < x 0 < p. We prove that |x 0| ≪ p 3/2 q −1 log p. By applying the Burgess character sum estimates, and estimates of certain exponential sums due to Bourgain, Glibichuk and Konyagin, we derive refinements of our result.