Intuitive Idea: Say we have polynomials modulo positive integers; for instance, $f(x) = 3x + 1\pmod 5$ and $g(x) = 5x^2 - x + 3 \pmod 7$. (Yes, there is some abuse of notation here.) We may "compose" these functions with each other. For instance, $f(3) = 9 + 1 \equiv 0 \pmod 5$, and $g(0) \equiv 5\cdot0 - 0 + 3 \equiv 3 \pmod 7$. We may write this as $(gf)(3) = 3$. Some other outputs are $(gf)(1)=g(4)=2$, $(gf)(2)=g(7) \equiv g(0)=3\pmod 7$, and so on. The question is, is this composition function $(gf)(x)$ the same as some $p(x) \pmod m$, where $p$ is a polynomial and $m$ is a positive integer?
Formalization with maps: For any integer n, let $i_n : \mathbb{Z}_p \mapsto \mathbb{Z}$ be the inclusion and let $h_n : \mathbb{Z} \mapsto \mathbb{Z}/n\mathbb{Z} = \mathbb{Z}_n$ be the surjective group homomorphism (is there a special name for this map?). Let $p,q \in \mathbb{N}$. For all $f \in \mathbb{Z}_p[x]$ ,$g \in \mathbb{Z}_q[x]$, does there exist some $r \in \mathbb{N},h \in \mathbb{Z}_r[x]$ such that $r = h_r(i_qgh_q)(i_pfh_p)i_r$?
