I'm having problems finding an appropriate reference for this question.

Given two elements $f, g \in \mathbb{C}[x_1, \dots, x_n]$, consider their greatest common divisor, $\gcd_{\mathbb{C}[x_1, \dots, x_n]}(f, g) = h \in \mathbb{C}[x_1, \dots, x_n]$.

Now, consider this two elements $f$ and $g$ as elements of $\mathbb{C}[[x_1, \dots x_n]]$, i.e., formal power series with finitely many terms. Compute $\gcd_{\mathbb{C}[[x_1, \dots, x_n]]}(f, g) = h' \in \mathbb{C}[[x_1, \dots, x_n]]$.

Does exists a unit $u \in \mathbb{C}[[x, y]]$ such that $h = uh'$? This is equivalent to asking if $h$ is a $\gcd$ in $\mathbb{C}[[x, y]]$.

I'm sure this can be rephrased in terms of irreducible components of algebraic vs. analytic varieties.

Thanks.