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

Let W(k,2) denote the least number s for which the system of equations Sigma(i=1)(s)x(f)(j)=Sigma(i=1)(s)y(i)(j)(1 less than or equal to j less than or equal to k) has a solution with Sigma(i=1)(s)x(i)(k+1)not equal Sigma(i=1)(s)y(i)(k+1). We show that for large k one has W(k, 2)less than or equal to 1/2k(2)(log k+log log k+O(1)), and moreover that when K is large, one has W(k,2)less than or equal to 1/2k(k+1)+1 for at least one value k in the interval [K,K-4/3 divided by epsilon]. We show also that the least s for which the expected asymptotic formula holds for the number of solutions of the above system of equations, inside a box, satisfies s less than or equal to k(2)(log k+O(log log k)).