I ask this question in response to a discussion on this question.
If $f(x)=p(x)/q(x)$ is a one-variable real rational function, then for each $r\in \mathbb{R},$ $f(x)=r$ has at most $d=\max\{\deg(p),\deg(q)\}$ solutions; this is due to the Fundamental Theorem of Algebra. So, for instance, if we had a rational function of one variable of the form quadratic/quadratic that took the same value at four points, we could conclude it is constant.
Does a similar result hold for multivariable functions? If so, when; if not, does a partial result hold?
