Find Parameter Set for N Intersections of 2 Implicit Functions

[Dovetailed]

Sorry for the wordiness of the thread title.

Basically I'm wondering, if you have two implicit functions, F(x,y)=0 and G(x,y)=0 (typically rational functions with numerator and denominator very high degree polynomials), both dependent upon the same K (in my case > 34) dimensional set of parameters - a for i = 1 to K; are there any established numerical techniques to speed up a search of the parameter space to find a set of parameters for which the 2 curves in the x-y plane (defined by the functions F and G) have a given number of intersections, say N?

Cheers!

 

[fresh_42]

You can apply algebraic geometry to some extent, but this depends on many things you haven't said. But algebraic geometry would be the direction in which I sought.