Integral solutions to systems of diagonal equations

TL;DR

This work analyzes diagonal systems $M \boldsymbol{x}^d=\boldsymbol{\mu}$ and obtains asymptotic counts of integer solutions under a rank-type condition $\Psi(M) \ge T_{\text{int.}}(d)+1$, with a matching result for solutions where variables are restricted to $Z$-smooth numbers under $\Psi(M) \ge T_{\text{smo.}}(d)+1$. The authors implement the Hardy–Littlewood circle method, leveraging state-of-the-art Weyl sum bounds from the resolution of Vinogradov’s mean value theorem and Waring’s problem progress to control minor arcs and extract a main term given by the product of a singular series $\mathfrak{S}$ and a singular integral $\mathfrak{I}$. They establish the convergence and evaluation of $\mathfrak{S}$ and $\mathfrak{I}$, show that the leading constant is positive under local solubility, and provide explicit thresholds $T_{\text{int.}}(d)$ and $T_{\text{smo.}}(d)$ that determine when the asymptotics hold. The results advance previous variable-count requirements for diagonal systems and demonstrate the power of combining recent number-theoretic bounds with the circle method for Diophantine geometry.

Abstract

In this paper, we obtain an asymptotic formula for the number of integral solutions to a system of diagonal equations. We obtain an asymptotic formula for the number of solutions with variables restricted to smooth numbers as well. We improve the required number of variables compared to previous results by incorporating the recent progress on Waring's problem and the resolution of the main conjecture in Vinogradov's mean value theorem.

Theorems & Definitions (38)

• Theorem 1.1: Davenport and Lewis
• Definition 1.2
• Remark 1.3
• Theorem 1.4
• Theorem 1.5
• Remark 1.6
• Corollary 1.7
• Lemma 2.1
• proof
• Lemma 2.2