Let $p$ be a prime number and $q=p^n$ for some positive integer $n$. $F_q[x]$ is the polynomial ring with coefficients in $F_q$. For any $M(x)\in F_q[x]$, define $\mathcal{R}(M(x))\subset F_q[x]$ to be the set of all polynomials which has degree less than $M(x)$.
Fix $P(x)\in F_q[x]$, is there a sequence of functions $\{M_n(x)\}_{n\geq1}\subset F_q[x]$ such that $$\left\{P^i(x)(mod ~M_n(x)): i\in\mathbb{N}\right\}=\mathcal{R}(M(x))$$ for every $n\geq1?$
Fix $P(x),Q(x)\in F_q[x],(P(x),Q(x))=1$, is there a sequence of functions $\{M_n(x)\}_{n\geq1}\subset F_q[x]$ such that $$\left\{P^i(x)(mod ~M_n(x)): i\in\mathbb{N}\right\}=\mathcal{R}(M(x))$$ and $$\left\{Q^j(x)(mod ~M_n(x)): j\in\mathbb{N}\right\}=\mathcal{R}(M(x))$$ for every $n\geq1?$
I think this is about primitive element of finite fields. and I know that if
$P(x)=x$ (corollary2.3)
then the first quenstion is valid. But is the general case right?