BOUNDS ON EXPONENTIAL SUMS AND THE POLYNOMIAL WARING PROBLEM MOD p

Abstract

Estimates are given for the exponential s u m ∑ x = 1 p exp ( 2 π i f ( x ) / p ) , p a prime and f a nonzero integer polynomial, of interest in cases where the Weil bound is worse than trivial. The results extend those of Konyagin for monomials to a general polynomial. Such bounds readily yield estimates for the corresponding polynomial Waring problem mod p, namely the smallest γ such that f(x1)+…+f(xγ)≡N (mod p) is solvable in integers for any N.