Coprime polynomials and polynomial substitution

Question

Let $F$ be a field, and let $P(X_1,\dots,X_m)$, $Q(X_1,\dots,X_m) \in F[X_1,\dots,X_m]$ be two coprime polynomials. Consider $n$ new polynomials $R_1(Y_{1,1},\dots,Y_{1,n}) \in F[Y_{1,1},\dots,Y_{1,n}]$, $\dots$, $R_m(Y_{m,1},\dots,Y_{m,n}) \in F[Y_{m,1},\dots,Y_{m,n}]$ in the new $mn$ indeterminates $Y_{1,1},\dots,Y_{1,n},\dots,Y_{m,1},\dots,Y_{m,n}$, and assume that $R_1,\dots,R_m$ all have positive degree. My question is: are the polynomials $P(R_1(Y_{1,1},\dots,Y_{1,n}),\dots,R_m(Y_{m,1},\dots,Y_{m,n}) )$ and $Q(R_1(Y_{1, 1},\dots,Y_{1, n}),\dots,R_m(Y_{m,1},\dots,Y_{m,n}))$ coprime?

I don't know the answer in general, neither I can say whether it is a trivial question or a deep one. Does someone have some idea about it? Thank you very very very much in advance for your kind attention.

NOTE 1 The answer is positive for $m=1$. This is an easy consequence of Bézout's Identity, by which there exist $S(X),T(X) \in F[X]$ such that \begin{equation} S(X)P(X)+T(X)Q(X)=1, \end{equation} so that for any $R(Y_1,\dots,Y_n) \in F[Y_1,\dots,Y_n]$ we have \begin{equation} S(R(Y_1,\dots,Y_n))P(R(Y_1,\dots,Y_n))+T(R(Y_1,\dots,Y_n))Q(R(Y_1,\dots,Y_n))=1, \end{equation} which implies that $P(R(Y_1,\dots,Y_n))$ and $Q(R(Y_1,\dots,Y_n))$ are coprime. Note that this results holds true even if $R(Y_1,\dots,Y_n)$ is a constant polynomial. For $m > 1$ we cannot allow $R_1,\dots,R_m$ to be constant polynomials, otherwise the answer is for sure in general negative: take e.g. $P(X_1,X_2)=X_1+X_2$, $Q(X_1,X_2)=X_1$, $R_1(Y_{1,1},Y_{1,2})=Y_{1,1}$,$R_2(Y_{2,1},Y_{2,2})=0$.

NOTE 2 This question has been posted on math.stackechange.com as Substituting Polynomials into Two Relatively Prime Polynomials, but it has received no attention there, whether it has been deemed too difficult or of little interest.

Answer 1

The answer is positive. Since $P,Q$ are coprime the variety $V=\{P=Q=0\}\subset\mathbb A^m$ has dimension $\le m-2$. Consider the variety $W=\{P(R_1,\ldots,R_m)=Q(R_1,\ldots,R_m)=0\}\subset\mathbb A^{nm}$ and the map $f=(R_1,\ldots,R_m):W\to V$. The fibers of $f$ have dimension $m(n-1)$ (because $R_i$ are non-constant and each is applied to a separate set of variables) and therefore $\dim W=\dim V+m(n-1)\le m(n-1)+m-2=mn-2$, so $P(R_1,\ldots,R_m),Q(R_1,\ldots,R_m)$ are coprime.