Multiple Gauss sums

A multiple Gauss sum is a complete multiple exponential sum twisted by Dirichlet characters. We prove a new bound for multiple Gauss sums and, as an application, improve previous results in the Birch–Goldbach problem. Let $F_1,\dots ,F_R\in \mathbb{Z}[x_1,\dots ,x_s]$ be forms with differing degrees, with $D$ being the highest degree, and let $\mathit{F}=(F_1,\dots ,F_R)$ be nonsingular. We prove that the system $\mathit{F}(\mathit{x})=\mathbf{0}$ is solvable in primes provided that $s\ge D^2{4}^{D+2}{R}^5$.