BOUNDS ON EXPONENTIAL SUMS AND THE POLYNOMIAL WARING PROBLEM MOD p

Estimates are given for the exponential sum sigma{x=1}-p \exp(2\pi 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 gamma such that f(x_1)+... +f(x_{\gamma})\equiv N (mod p) is solvable in integers for any N