Subconvexity and the Hilbert-Kamke Problem

2022/01/07 by Wooley, Trevor D.
#11D72 #11L07 #11P55 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2201.02699

Abstract

When s≥ k≥ 3 and n₁,… ,n_k are large natural numbers, denote by A_s,k(\mathbf n) the number of solutions in non-negative integers \mathbf x to the system x₁^j+… +x_s^j=n_j (1≤ j≤ k). Under appropriate local solubility conditions on \mathbf n, we obtain an asymptotic formula for A_s,k(\mathbf n) when s≥ k(k+1). This establishes a local-global principle in the Hilbert-Kamke problem at the convexity barrier. Our arguments involve minor arc estimates going beyond square-root cancellation.

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