# Solution of polynomial equations

Suppose there is a set of complex variables

$$\{x_i,\,i=1 \ldots M;\;\;y_k,\,k=1 \ldots N\}$$

and a polynomial equation

$$p(x_i, y_k) = 0$$

Is there a way to prove or disprove for such an equation whether it can be reformulated as

$$f(x_i) = g(y_k)$$

with two functions f and g with

$$\nabla_y f= 0$$
$$\nabla_x g= 0$$

Hey tom.stoer.

I'm not familiar with the polynomial constraint you use: can you point me to some more specific definition? (I'm sorry but I'm only familiar with a univariate polynomial).

You mean the gradient? It's only to stress that f does not depend on y1, y2, ... and g does not depend on x1, x2, ...

No sorry, I mean the p(x_i,y_i) = 0. Is this basically the product of univariate polynomials that is equal to 0?

No, its a general multivariate polynomial in xi and yi, no special construction, no special condition.