I have been working independently on a project but now I am stuck and need to seek an expert's wisdom for a part of it. I am basically looking for theorems related to growth of the size of polynomials. My Google search has led me to papers on root-finding algorithms which is not what I intend to do. 


I am dealing with polynomials $p(x)$ of degree at least $2$ in $\mathbb{Z}[x]$ with a non-zero constant term and the size of the $n$-th iteration, i.e. $|p^{\circ n}(a)|$, where $a$ is any integer. 


My example was very specific but I am looking for anything remotely similar. Is there any results that you know of related to the size and the growth rate of polynomials? Any comparison or upper/lower bound theorems?



Growth of the size of iterated polynomials
The quantity you're looking for, I think, is the *canonical height* of $a$ with respect to the polynomial $p(x)$. In your situation, since you're just dealing with integers, if we let $d=\deg(p)\ge 2$, then the canonical height is defined by
$$ \hat h\_p(a) = \lim\_{n\to\infty} \frac{1}{d^n}\log\bigl|p^n(a)\bigr|. $$
The fact that the limit converges is due to Tate, and a very general version was proven by Call and me. The canonical height has many nice properties, including the following:


* There is a constant $C=C(p)$ so $\bigl|\hat h\_p(b) - \log|b|\bigr|<C$ for all integers $b$.
* $\hat h\_p\bigl(p(b)\bigr) = d \hat h\_p(b)$ for all integers $b$.
* The integer $b$ is preperiodic for $p$ if and only if $\hat h\_p(b)=0$. (Being preperiodic means that there are $m>n\ge0$ so that $p^m(b)=p^n(b)$. Note that in any case we have $\hat h\_p(b)\ge0$.)


In particular, if your $a$ is not preperiodic, then using these facts you get a pretty good estimate for $\log\bigl|p^n(a)\bigr|$, namely
$$
 \log\bigl|p^n(a)\bigr| = \hat h\_p\bigl(p^n(a)\bigr) + O(1)
 = d^n \hat h\_p(a) + O(1),
$$
where the $O(1)$ depends on $p$, but not on $a$ or $n$. Further, with more work, one can even show that if $B$ is a bound for the largest coefficient of $p(x)$, then the $O(1)$ can be replaced by $O(\log|B|)$, where now the big-$O$ constant depends only on the degree of $p(x)$. Here are a couple references for all of this:


Section 3.4 of *The Arithmetic of Dynamical Systems*, J.H. Silverman, Springer, GTM 241.


G. Call, J.H. Silverman, Canonical heights on varieties with
morphisms, *Compositio Math.* **89** (1993), 163-205.


