Subconvexity and the Hilbert-Kamke Problem

TL;DR

This paper establishes an asymptotic formula for the number of solutions to a system of Diophantine equations related to the Hilbert-Kamke problem, demonstrating a local-global principle at the convexity barrier using advanced exponential sum estimates.

Contribution

It provides a new asymptotic formula for the Hilbert-Kamke problem when the number of variables exceeds the convexity barrier, advancing understanding of local-global principles in this context.

Findings

Established asymptotic formula for solution count when s ≥ k(k+1)

Demonstrated local-global principle at the convexity barrier

Developed minor arc estimates beyond square-root cancellation

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.