Let $P_1(X),P_2(X),Q_1(X),Q_2(X)\in\overline{\mathbb{Q}}[X]$ (or $\mathbb{Q}[X]$) and $r(X)\in\overline{\mathbb{Q}}(X)$ (or $\mathbb{Q}(X)$).
Let $\frac{P_1(r(x))}{Q_1(r(x))}=\frac{P_2(x)}{Q_2(x)}$.
Let's consider coprimality over $\overline{\mathbb{Q}}$ (or $\mathbb{Q}$).
1.)
In which cases can the coprimality of $P_2(x)$ and $Q_2(x)$ be concluded from the coprimality of $P_1(x)$ and $Q_1(x)$?
2.)
In which cases can the coprimality of $P_1(x)$ and $Q_1(x)$ be concluded from the coprimality of $P_2(x)$ and $Q_2(x)$?
3.) Is this possible at least for some kinds of $r(X\in\overline{\mathbb{Q}}[X]$ (or $\mathbb{Q}[X]$)?
I don't know how I could start to derive an answer.
The question is important for answering the question when an irreducible rational equation keeps irreducible after inserting a rational expression.
I need the answer because I want to try to prove some of the conjectures in Closed-form solubility of elementary transcendental equations?.
Both answers are "always", at least when $r$ is nonconstant.
(1): If $P_2(x)$ and $Q_2(x)$ have the factor $G(x)$ in common, then let $\alpha$ be a root of $G(x)$ in $\overline{\Bbb Q}$. We have $P_2(\alpha)=Q_2(\alpha)=0$, and so if $\beta=r(\alpha)$, we have $P_1(\beta)=Q_1(\beta)=0$. Then if $H(x)$ is the minimal polynomial of $\beta$ (over whatever field we're working over), we conclude that $P_1(x)$ and $Q_1(x)$ have the factor $H(x)$ in common.
(2): If $P_1(x)$ and $Q_1(x)$ have the factor $F(x)$ in common, then $P_2(x)$ and $Q_2(x)$ will have the factor $F(r(x))$ in common.