Is there a notion of transcendental numbers for polynomials of multiple variables?

Question

We say $\alpha\in\mathbb{C}$ is algebraic if there is a polynomial $p(x)\in\mathbb{Z}[x]$ such that $p(\alpha)=0$, and we say $\alpha$ is transcendental if no such polynomial exists. Examples of transcendental numbers include $\pi,e,\sum_{n\ge0}10^{-n!}$, etc. But this definition applies only to polynomials of one variable: is there a notion of transcendental numbers for polynomials of multiple variables? I'm unsure how it would look: for instance, $p_1=\pi$ and $p_2=\pi+1$ are both transcendental but they satisfy $P(x,y) = x-y+1$. If there is such a generalization, is there a set $S$ of numbers that are "totally transcendental," in the sense that any finite subset of $S$ doesn't satisfy a polynomial with integer coefficients in the corresponding number of variables?