For example, are there known bounds on the roots w/r to the $p$-adic metric? Ways to determine how many roots are in $\mathbb{Q}$, or in $\mathbb{Q}_p$ itself as opposed to an algebraic extension?
Specifically, I am considering the following problem: given the Collatz polynomial polynomial $P_N$ whose constant term is $N$, whose leading coefficient is presumed to be 1, and such that $a_{k+1}$ is the Collatz iterate of $a_k$ if $a_k\neq 1$, what do the roots of $P_N$ look like? Using Newton polygons I have managed to prove that every odd $N$ other than powers of $3$ has a root in $\mathbb{Z}_p$ for some odd $p\neq 3$ dividing $N$ and that this root has valuation $m$ where $p^m\mid\mid N$, but am not sure what else to do with this information.
Possibly relevant is that these polynomials are linear combinations in $\mathbb{Z}[X]$ of products of cyclotomic polynomials; if some $\Phi_d$ appears in every product then the polynomial is reducible, but the converse is not true (e.g., $P_5(-1) = 0$). I know that the $n^{th}$ root of unity $\omega_n$ is in $\mathbb{Q}_p$ iff $n\mid p-1$.