Structure and paucity in affine diagonal systems, I

Let $\epsilon >0$ and $\mathbf{h}\in {\mathbb{Z}}^3$. We show that whenever $P$ is large and the system $$x_1^j+x_2^j-y_1^j-y_2^j=h_j(j=1,2,3)$$ has more than $P^{\epsilon }$ 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.