A variant of the Schwartz–Zippel lemma Let $f \in \mathbb{F}[x_1,\ldots,x_n]$ be a nonzero polynomial. Let $d_1$ be the maximum exponent of $x_1$ in $f$ and let $f_1$ be the coefficient of $x_1^{d_1}$ in $f.$ Let $d_2$ be the maximal exponent of $x_2$ in $f_1$ and so on for $d_3,\ldots,d_n.$
I would like to show that if $S_1, \ldots, S_n \subseteq \mathbb{F}$ are arbitrary subsets and $r_i \in S_i$ are chosen uniformly at random then $$ Pr[f(r_1,\ldots,r_n) = 0] \leq \frac{d_1}{|S_1|} + \cdots + \frac{d_n}{|S_n|}.$$
The claim is obviously true for $n = 1$ but for multivariate polynomials I don't see how to prove the claim since I don't see how to handle the fact that we disregard from $f$ all the terms containing $x_1^{k}$ for $k < d_1.$

This "variant of the Schwartz-Zippel Lemma" is in fact Lemma 1 from Jack Schwartz's original paper.
Note that although it looks nice to phrase the statement and/or the proof in probabilistic language, it is certainly not necessary to do so: Schwartz phrases it as a pure counting argument, which is (even) shorter than Jernej's (nice) solution above.  I mention this because the first time I saw this kind of proof of the Schwartz-Zippel Lemma, I think I took the probabilistic stuff a bit too seriously: it really is a way of phrasing the argument rather than a proof technique.   
I hope the textbook gives some indication of the provenance of this "exercise"!
