Physical background:
In the Bargmann-Segal representation, the states of bosonic quantum systems are holomorphic functions of $N$ complex variables where $N$ is the number of degrees of freedom. The states belong to a Hilbert space with the scalar product defined with the help of the measure $\exp(-\sum_{i}{z_{i}^{\star} z_{i}}) d z_{1}... d z_{N}$.
Now, the question is: is it possible to retrieve a quantum state from its nodes? That is, is it possible to retrieve a holomorphic (entire) function of $N$ variables from its manifold of zeros?
As far as I understand, in the case of $N = 1$ we have the Weierstrass factorization theorem which gives a positive answer. So, does there exist a generalization of that theorem to $N > 1$?
To be even more ambitious: let us suppose we have a curve in $\Bbb R^{3}$, a torus knot, say. Given the information that it is an intersection of a complex curve $K$ with a sphere, and the form of the knot, can we retrieve the entire function $f(z_{1}, z_{2})$ such that $f(z_{1}, z_{2}) = 0$ is an equation of $K$?
I apologize if the question is naive, and imprecise; I'm a physicist, not a mathematician. And thank you for your attention.