  We obtain a new bound for incomplete Gauss sums modulo primes. Our argument
falls under the framework of Vinogradov's method which we use to reduce the
problem under consideration to bounding the number of solutions to two distinct
systems of congruences. The first is related to Vinogradov's mean value
theorem, although the second does not appear to have been considered before.
Our bound improves on current results in the range $N\ge
q^{2k^{-1/2}+O(k^{-3/2})}$.
