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?