Is there a "Bezout's theorem" for analytic curves? Let $\varphi_1(u,v)$ and  $\varphi_2(u,v)$ be two entire or meromorphic functions in the two complex variables $u$ and $v$.  If they are both polynomials, then Bezout's Theorem says that the set of common roots $(u,v)$ of $\varphi_1(u,v)=0$ and $\varphi_2(u,v)=0$ is,  with suitable qualifications, a finite set whose cardinality is the product of the degrees of $\varphi_1(u,v)$ and  $\varphi_2(u,v)$.  There is a similar statement for the common roots of three polynomial equations in three variables, and so on.
Here is my question:  Is there a corresponding theorem for the case when $\varphi_1(u,v)$ and  $\varphi_2(u,v)$ are transcendental functions, i.e., in general is their common set of roots an infinite discrete point set in two-dimensional complex space?
If one is transcendental and the other is linear, then the result is true...take the case of $v=\sin u$ and $v=0$.  Picard's theorem gives a general answer in that case.  But, what happens when both functions are transcendental?
 A: 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.
A: sin(x) and x + sin(x) are both transcendental right? 
A: I am not an expert but I'd like to add that there is a theory called Value Distribution Theory (or Nevanlinna Theory) which counts how many elements there are in your set within a ball of radius $R$ and the behaviour as $R \to \infty$. In particular, something like the "second main theorem with moving targets" should give what you want at least when your analytic curves $\phi = 0$ are graphs.
