Analog to the identity theorem for holomorphic functions for functions of two complex variables

Question

Consider a function $f:\mathbb{C}^2\rightarrow\mathbb{C}$ that is holomorphic and vanishes on a non empty bounded open set $S$. If $f$ were a function from $\mathbb{C}$, by the identity theorem, $f$ would be uniquely determined to be the zero function. Does a similar theorem hold when the domain is $\mathbb{C}^2$? If not, how can the vector space of all functions vanishing on $S$ be categorized or expressed? For example, I would be interested in listing them all if possible, or knowing the dimension of the vector space of all such functions.

Answer 1

Sure, you can deduce this from the single-variable case. Pick a point $p\in S$, so $f$ vanishes on a neighborhood of $p$. Now for any other $q\in\mathbb{C}^2$, consider the function $g(z)=f(zq+(1-z)p)$. This is a holomorphic function $\mathbb{C}\to\mathbb{C}$, and is basically the restriction of $f$ to the (complex) line between $p$ and $q$. Since $f$ vanishes in a neighborhood of $p$, $g$ vanishes in a neighborhood of $0$, so $g=0$. In particular, $g(1)=0$ so $f(q)=0$. Since $q$ was arbitrary, $f=0$.