How many evaluations do you need to prove that a multivariate polynomial is the zero polynomial?

Question

For a univariate polynomial $f$, you just need to prove that $f(x) = 0$ for $d+1$ distinct $x$ to prove that $f$ is the zero polynomial.

But for multivariate polynomials, how does that work? How many evaluations do you need to fully describe a multivariate polynomial?