ca.analysis and odes - Is there a "Bezout's theorem" for analytic curves?

I am sure that the theorem you want is true, but I am missing one technical reference.

Claimed Theorem: Let $f(x,y)$ and $g(x,y)$ be entire functions on $mathbb{C}^2$. Then either ${ f=0 }$ and ${ g=0 }$ have a one dimensional overlap, or ${ f=g=0 }$ is discrete.

"Proof": If ${ f=g=0 }$ is not discrete, then it has an accumulation point $z$. Suppose ${ f=0 }$ is smooth at $z$. Then, by the implicit function theorem, we can locally parameterize ${ f=0 }$ as ${ (a(t), b(t)): t in D }$ where $D$ is a small disc and $a$ and $b$ are alanlytic functions on $D$, with $(a(0), b(0))=z$. Then the function $g(a(t), b(t))$ has infinitely many zeroes with an accumulation point at $t=0$, so it must be identically zero. Thus, $g$ vanishes on an open set in ${f=0 }$.` QED

Now, the gap in the above is that $z$ might not be a smooth point.

What I am quite certain is true, but I don't know a reference for, is that we have resolution of singularities for analytic germs the same way we do for polynomials. So, for any nonzero analytic function $f(x,y)$ and any point $z$, there is a neighborhood of $z$ where we can factor $f$ as $f_1 f_2 cdots f_r$, each an analytic function, and such that ${ f_i=0 }$, near $z$, can be paratemerized ${ (a_i(t), b_i(t)): t in D }$. One can probably prove this by brute force, but I'm sure one of our readers knows a reference.