a1 Department of Mathematics, University of Michigan Ann Arbor, MI 48109-1003 U.S.A..
The object of this paper is to obtain improvements in Vinogradov's mean value theorem widely applicable in additive number theory. Let Js,k(P) denote the number of solutions of the simultaneous diophantine equations
with 1 ≥ xi, yi ≥ P for 1 ≥ i ≥ s. In the mid-thirties Vinogradov developed a new method (now known as Vinogradov's mean value theorem) which enabled him to obtain fairly strong bounds for Js,k(P). On writing
in which e(α) denotes e2πiα, we observe that
where Tk denotes the k-dimensional unit cube, and α = (α1,…,αk).
(Received September 09 1990)