For a function of a single variable, the maximum number of critical points is just equal to the order of the polynomial -1. So a fourth degree polynomial function has a maximum of 3 critical points.

I was wondering what the case would be for a function of n variables. Let's say you have a function f(x,y) = x^3 + y^3 + ..... etc

I have a feeling the maximum number of critical points should be 4, or just the highest powers of x and y -1 added together. I don't know if it's right, or how to prove it though...