On the inhomogeneous Vinogradov system

TL;DR

This paper investigates the number of solutions to an inhomogeneous Vinogradov system, showing it has fewer solutions in certain ranges compared to the homogeneous case, using advanced mean value theorems and shifting techniques.

Contribution

It introduces new bounds on solutions for the inhomogeneous Vinogradov system in the subcritical range, extending previous homogeneous results.

Findings

Fewer solutions in the inhomogeneous case than the homogeneous case in certain ranges.

Uses Vinogradov's mean value theorem with a shifting argument to derive bounds.

Provides quantitative estimates for the solution count in the inhomogeneous system.

Abstract

We show that the system of equations \begin{align*} \sum_{i=1}^s (x_i^j-y_i^j) = a_j \qquad (1 \le j \le k) \end{align*} has appreciably fewer solutions in the subcritical range $s < k(k+1)/2$ than its homogeneous counterpart, provided that $a_\ell \neq 0$ for some $\ell \le k-1$. Our methods use Vinogradov's mean value theorem in combination with a shifting argument.