Structure and paucity in affine diagonal systems, I

TL;DR

This paper investigates the conditions under which certain affine diagonal systems have many or few integral solutions, revealing a dichotomy between solution scarcity and structural constraints on parameters.

Contribution

It establishes a clear criterion linking the abundance of solutions to the algebraic structure of the system's coefficients, advancing understanding of solution distribution in Diophantine equations.

Findings

If the system has more than P^ε solutions, then the coefficients are highly structured.

The system exhibits a dichotomy: either solutions are scarce or coefficients follow a specific form.

Implications for higher-variable systems and related paucity problems.

Abstract

Let $\varepsilon>0$ and $\mathbf h\in \mathbb Z^3$. We show that whenever $P$ is large and the system \[ x_1^j+x_2^j-y_1^j-y_2^j=h_j\quad (j=1,2,3) \] has more than $P^\varepsilon$ integral solutions with $1\le x_i,y_i\le P$, then there exist natural numbers $a$ and $b$ with $h_j=a^j-b^j$ $(j=1,2,3)$. This example illustrates the theme that, either the Diophantine system has a paucity of integral solutions, or else the coefficient tuple $\mathbf h$ is highly structured. We examine related paucity problems as well as some consequences for problems involving more variables.