Coprime multivariate polynomials Let ${\bf R}$ be a gcd domain, $n \geq 2$, $k \in \mathbb{N}^*$, and $f,g \in 
{\bf R}[X_1,\ldots,X_n]$. Supposing that $f$ and $g$ are coprime in ${\bf R}[X_1,\ldots,X_n]$, that is, $\gcd(f,g)=1$, can we say that $f(X_1^k,\ldots,X_n^k)$ and $g(X_1^k,\ldots,X_n^k)$ are also coprime in ${\bf R}[X_1,\ldots,X_n]$?
 A: The answer is yes.

Claim. Let $R$ be an integral domain and let $f(X_1, \dots, X_n), g(X_1, \dots, X_n) \in R[X_1, \dots, X_n]$ $(n \ge 1)$ be polynomials with no non-constant common divisor. Then $f(X_1^k, \dots, X_n^k)$ and $g(X_1^k, \dots, X_n^k)$ have no non-constant common divisor for every $k \ge 1$.

The result follows since a polynomial of degree zero divides $f(X_1, \dots, X_n)$ in $R[X_1, \dots, X_n]$ if and only if it divides $f(X_1^k, \dots, X_n^k)$ for some (equivalently, for all) $k \ge 1$.

Proof. Assume first that $n = 1$ and let $K$ be the field of fractions of $R$. Since $K[X]$ is a principal ideal domain, we can find $p(X), q(X) \in R[X]$ and $r \in R$ such that $p(X)f(X) + q(X)g(X) = r$ and hence $p(X^k)f(X^k) + q(X^k)g(X^k) = r$. Thus $f(X^k)$ and $g(X^k)$ have no common divisor of positive degree.
Suppose now that $n > 1$.
Assume moreover that we can find $i \in \{1, \dots, n \}$ such that both $f(X_1, \dots, X_n)$ and $g(X_1, \dots, X_n)$ have positive degree with respect to $X_i$. Resorting to the case $n = 1$, we see that a common divisor of $f(X_1^k, \dots, X_n^k)$ and $g(X_1^k, \dots, X_n^k)$ has degree zero with respect to $X_i$. Hence such a divisor has total degree zero.
If there is no integer $i$ as above, i.e., if $f$ and $g$ depend on two disjoint sets of variables, the result is then obvious.

