Subconvexity and the Hilbert-Kamke Problem

Abstract

When $s\ge k\ge 3$ and $n_1,\dots ,n_k$ are large natural numbers, denote by $A_{s,k}(\mathbf{n})$ the number of solutions in non-negative integers $\mathbf{x}$ to the system \[ x_1^j+\ldots +x_s^j=n_j\quad (1\le j\le k). \] Under appropriate local solubility conditions on $\mathbf{n}$, we obtain an asymptotic formula for $A_{s,k}(\mathbf{n})$ when $s\ge k(k+1)$. 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.

arXiv comment13 pages