Strong paucity in the Br\"udern-Robert Diophantine system

TL;DR

This paper proves a significant scarcity result for solutions of a specific Br"udern-Robert Diophantine system, showing that non-diagonal solutions are much fewer than diagonal ones, with an improved upper bound.

Contribution

It establishes a stronger paucity estimate for solutions of the Br"udern-Robert system, improving previous bounds and demonstrating the rarity of non-diagonal solutions.

Findings

Non-diagonal solutions are significantly fewer than diagonal solutions.

The difference between total and diagonal solutions is bounded by P^{√(8k+9)-1+ε}.

The result improves earlier bounds on solution scarcity.

Abstract

Let $k$ be a natural number with $k\ge 2$, and let $\varepsilon>0$. We consider the number $V_k^*(P)$ of integral solutions of the system of simultaneous Diophantine equations \[ x_1^{2j-1}+\ldots +x_{k+1}^{2j-1}=y_1^{2j-1}+\ldots +y_{k+1}^{2j-1}\quad (1\le j\le k), \] with $1\le x_i,y_i\le P$ $(1\le i\le k+1)$. Writing $L_k^*(P)$ for the number of diagonal solutions with $\{x_1,\ldots ,x_{k+1}\}=\{y_1,\ldots ,y_{k+1}\}$, so that $L_k^*(P)\sim (k+1)!P^{k+1}$, we prove that \[ V_k^*(P)-L_k^*(P)\ll P^{\sqrt{8k+9}-1+\varepsilon}. \] This establishes a strong paucity result improving on earlier work of Br\"udern and Robert.