For systems of equations without diagonal structure, the circle method typically calls for a number of variables which is exponential in the degree and quadratic in the number of equations. In the diagonal case, conclusions which are polynomial in the degree and linear in the number of equations have long been known thanks to work on Vinogradov's mean value theorem. We describe how to attain a number of variables which is linear in the number of equations, for a smooth system of not-necessarily-diagonal equations of the same degree.