It is well know fact that:
A polynomial $f$ (in many variables) and its derivative $\partial_x f$ (with respect to $x$) have common roots (with respect to $x$) if and only if the discriminant of $f$ (wrt. $x$) is zero.
In the application I have, $f$ is not a polynomial but it is somewhat similar to one. To illustrate, an example is: $$f(x,y)=1-c(1-x+y)-c(x)$$ where $$c(z)=z^k\quad\text{with}\quad k\in \mathbb Q_{+}\text{ and } k>1$$ In the general case, $f$ is an linear combination of the function $c(\cdot)$ (plus some constant) evaluated at linear combinations of the variables plus constants.
My question is if there is any known way of generalizing the tools of algebraic geometry to cover such functions? More precisely, is there some way to extend the notion of resultants/discriminant such that the result quoted above holds?