Let $m$ be a positive integer. For $k = 1, 2, 3, ... m$, fix $g_k(x_1, ..., x_{k + 1}) \in \mathbb{Z}[x_1, ..., x_{k + 1}]$.
For any polynomial $p(x) \in \mathbb{Z}[x]$, let
$P_0(x) = p(x), P_1(x) = p(g_1(x, P_0(x)))$ and define inductively a "reflexive composition of $p$, of depth $n$, determined by $g_1, ..., g_n$" to be
$$P_{n}(x) = p(g_n(x, P_0(x), P_1(x), ..., P_{n - 1}(x))).$$
One may wonder when all reflexive compositions of depth $m$, determined by $g_1, ..., g_m$, are reducible. For instance, when $m = 1$, it follows from Hilbert's irreducibility theorem that $g_1$ must be of the form $x_1 + x_2G(x_1, x_2)$ for some nonzero $G(x_1, x_2) \in \mathbb{Z}[x_1, x_2]$.
I would appreciate a reference to any result in this direction or to a discussion of some related problem.
Thank you. Albertas