We apply the efficient congruencing method to estimate Vinogradov's integral for moments of order 2s, with 1 = k(2) - 1. In this way we come halfway to proving the main conjecture in two different directions. There are consequences for estimates of Weyl type and in several allied applications. Thus, for example, the anticipated asymptotic formula in Waring's problem is established for sums of s kth powers of natural numbers whenever s >= 2k(2) - 2k - 8 (k >= 6).
- WARING PROBLEM