Some remarks on Vinogradov's mean value theorem and Tarry's problem

LetW(k, 2) denote the, least numbers for which the system of equations has a solution with. We show that for largek one hasW(k, 2)1/2k²(logk+loglogk+O(1)), and moreover that whenK is large, one hasW(k, 2)1/2k(k+1)+1 for at least one valuek in the interval [K, K^3/4+]. We show also that the leasts for which the expected asymptotic formula holds for the number of solutions of the above system of equations, inside a box, satisfiessk²(logk+O(loglogk).Research supported in part by NSF grant DMS-9303505, an Alfred P. Sloan Research Fellowship, and a Fellowship from the David and Lucile Packard Foundation.