I have a question concerning elliptic functions that maybe you can help me shed light on. I am a theoretical physicist so I excuse myself preemptively for any mathematical imprecision.

It is a fact that any two elliptic functions $f(z)$ and $g(z)$, having the same periodicities, satisfy an algebraic relation $$F[f,g]=0\;,$$ where $F$ is a polynomial in two variables with constant coefficients.

Similarly, it is a fact that, given an elliptic function $f(z)$, its convolution with another generic function $h(z)$ is again an elliptic function, having the same periodicities: $$c(z) = \intop_{-\infty}^{\infty} dw\,f(z-w)h(w)\;,$$ $$\left\lbrace\begin{array}{c} f(z+\omega_1) = f(z) \\ f(z+\omega_3) = f(z)\end{array}\right.\quad \Longrightarrow \quad \left\lbrace\begin{array}{c} c(z+\omega_1) = c(z) \\ c(z+\omega_3) = c(z)\end{array}\right.\;.$$ The function $h(z)$ is such that the convolution is well-defined.

Given the two facts above, I deduce that there exist an algebraic relation of the form $$\mathcal{F}[f,c] = \sum_{n=0}^N\sum_{m=0}^M \alpha_{n,m} f(z)^n\,c(z)^m = 0\;.$$

My question is the following: is there a simple (or at the very least, feasible) way to find the integers $N$ and $M$ and, possibly, compute the coefficients $\alpha_{n,m}$?

Thanks a lot!