Some Remarks on Vinogradov's Mean Value Theorem and Tarry's Problem
نویسندگان
چکیده
Let W(k, 2) denote the least number s for which the system of equations ~ _ i x ~ = ~ S = l y i ( 1 <~j~k) has a solution with ~S=lx~+l v e ~ = l y ~ +1. We show that for large k one has W(k, 2) ~< 89 k + logtog k + O(1)), and moreover that when K is large, one has W(k, 2) ~< 89 + 1) + 1 for at least one value k in the interval [K, K 4/3 +~]. 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 ~< k2(log k + O(log log k)).
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