In (single-variable) complex analysis, given analytic functions $f$ and $g$ with $\gcd(f,g)=h$, one can find analytic functions $u$ and $v$ such that $uf+vg=h$. I’d like to know if the same holds in several variables; as a simple case, specifically,

Let $f,g\colon\mathbb{D}^2\to\mathbb{C}$ be analytic (in the bi-disc $\mathbb{D}^2$) with $h=\gcd(f,g)$. Does there exist analytic functions $u,v\colon \mathbb{D}^2\to\mathbb{C}$ such that $uf+vg=h$?

(Here $h:=\gcd(f,g)$ denotes an analytic function (in $\mathbb{D}^2$) such that $h(z_1,z_2)=0$ if and only if $f (z_1,z_2)=g(z_1,z_2)=0$).

In (single-variable) complex analysis, given analytic functions $f$ and $g$ with no common zeros, one can find analytic functions $u$ and $v$ such that $uf+vg=1$. I’d like to know if the same holds in several variables; as a simple case, specifically,

Let $f,g\colon\mathbb{D}^2\to\mathbb{C}$ be analytic (in the bi-disc $\mathbb{D}^2$) with no common zeros. Does there exist analytic functions $u,v\colon \mathbb{D}^2\to\mathbb{C}$ such that $uf+vg=1$?