Vinogradov systems with a slice off

2017/07/19 by Brandes, Julia, Wooley, Trevor D.
#11D45 #11L07 #11L15 #11P55 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1707.06047

Abstract

Let I_s,k,r(X) denote the number of integral solutions of the modified Vinogradov system of equations x₁^j+… +x_s^j=y₁^j+… +y_s^j (1≤ j≤ k, j≠ r), with 1≤ x_i,y_i≤ X (1≤ i≤ s). By exploiting sharp estimates for an auxiliary mean value, we obtain bounds for I_s,k,r(X) for 1≤ r≤ k-1. In particular, when s,k∈ \mathbb N satisfy k≥ 3 and 1≤ s≤ (k²-1)/2, we establish the essentially diagonal behaviour I_s,k,1(X)≪ X^s+ε.

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