Structure and paucity in affine diagonal systems, I

TL;DR

The paper analyzes how coefficient structure controls the number of integral solutions to affine diagonal Diophantine systems with many variables. It establishes a structure–paucity dichotomy: large solution counts force an explicit algebraic form on the coefficient vector $\mathbf h$, while otherwise the counts are small, quantified via $O(P^{\varepsilon})$ or near-square-root bounds. The authors treat three families—the affine cubic Vinogradov system, a relative system with exponents $\{1,2,4\}$, and Brüdern–Robert-type systems $U_{s,k}$—and deploy polynomial identities together with divisor bounds to obtain sharp upper bounds and precise structure theorems; conversely, when the structured form $h_j = a^j - b^j$ (or sums thereof) exists, they derive explicit counting asymptotics. The results extend to higher variables and yield bounds beyond the square-root barrier, clarifying when diagonally structured coefficients yield significantly larger counts. These insights illuminate the interplay between diagonal structure and solution scarcity in high-dimensional Diophantine systems.

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.

Theorems & Definitions (20)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Corollary 1.4
• Theorem 1.5
• Theorem 1.6
• Corollary 1.7
• Lemma 2.1
• proof
• proof : Proof of Theorem \ref{['theorem1.1']}