# algebraic number theory – If \$a_{1}=f(x)\$ and \$a_{r}=f(a_{r-1})\$ for \$r>1\$ then is it true that \$\$limsup_{rtoinfty}gamma(a_{r})rightarrowinfty?\$\$

Let $$f(x)$$ be a polynomial with integral coefficients and $$deg(f)geq 2$$.we define $$gamma(g(x))$$ to be the degree of the polynomial $$r(x)$$ which divides $$g(x)$$ and also has minimal degree.

For $$rgeq 1$$, We define the sequence $$a_{r}$$ of polynomials as follows:

$$clubsuit)a_{1}=f(x)$$

$$clubsuit)a_{r}=f(a_{r-1})$$ for $$rgeq 2$$

Then is it true that
$$limsup_{rtoinfty}gamma(a_{r})rightarrowinfty?$$