Subconvexity and the Hilbert-Kamke Problem

Trevor D. Wooley

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arXiv · 2201.02699

Subconvexity and the Hilbert-Kamke Problem

Trevor D. Wooley

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Abstract

When $s\ge k\ge 3$ and $n_1,\ldots ,n_k$ 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.

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BibTeX RIS

Trevor D. Wooley. 2022-01-07. Subconvexity and the Hilbert-Kamke Problem. https://arxiv.org/abs/2201.02699

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