Weyl's Inequality and Hua's Inequality over Number Fields

Abstract

We extend the previous work of Weyl’s inequality and Hua’s inequality over the real numbers to obtain a bound over general number fields L with [L : Q] = d. This produces two results: a bound on exponential sums P |x|<P e(tr(f(x))) ≪ Pd− 1 K +ε with K = 2deg(f)−1, and a bound of P(2m+1−(m+1))d+ε for the number of solutions to the equation xk1 +xk2 +...+xk2m = yk 1 +yk 2 +...+yk 2m, with |xi|, |yi| < P.