The set of all $p \in \mathbb{C}[x]$ that can be expressed using only one occurrence of $x$. Let $X$ denote the least subset of $\mathbb{C}[x]$ subject to the following constraints.


*

*$x \in X$.

*$p \in X \rightarrow ap \in X,$ for all $a \in \mathbb{C}$.

*$p \in X \rightarrow p+a \in X,$ for all $a \in \mathbb{C}$.

*$p \in X \rightarrow p^n \in X$, for all natural $n \geq 0$.
More conceptually, $X$ is the set of all $p \in \mathbb{C}[x]$ that can be expressed using at most one occurrence of $x$. For example, the polynomial $p = 18 x^6-12 x^3+7$ can be written as $2(3x^3-1)^2+5,$ hence $p \in X.$ 

Question. Can $X$ be described more explicitly? A little bit more precisely, given a polynomial $p = a_n x^n + \cdots + a_0 x^0$, is there an explicit condition (or algorithm) involving only the $a_i$'s that allows us to tell whether or not $p \in X$?

 A: Linear factors can always be pulled out, e.g. $a\cdot p^n = (\sqrt[n] a \cdot p)^n$. Thus it suffices to consider the subset $X_0\subset X$ of monic polynomials and try to determine if a given monic polynomial $p$ is in $X_0$ or not.
Let $p$ be a monic polynomial of degree $>1$.
If $p\in X_0$, then $p=q^m+c$ for some $q\in X_0$, $c\in\mathbb C$, $m\ge 2$.
We can try to find all candidates for $q$ and recursively test if they are $\in X_0$.
To test for the case $m=2$, not that in this case $p'=2q$; therefore we check whether $p-\frac14p'^2$ is constant (of course this immediately fails if $\deg p\ne 2$) and if so we have found a first candidate $q=\frac12p'$.
More generally, we note that $m$ must be a divisor of $\deg p$. Hence for each of these finitely many $m$ with $r:=\frac {n}m$, make an ansatz $q=x^r+b_{r-1}x^{r-1}+\ldots+b_0$ and solve for the $b_i$ from top downwads; we obtain $b_{r-i}$ from comparing coefficients of $x^{n-i}$, which gives us a linear equation, i.e. a unique solution for $b_{r-i}$. Once $q$ is found, check if $p-q^m$ is constant and if so, we have another candidate.
