Schwartz-Zippel lemma question
If $f = \prod_{i=1}^d x_i$ is a single monomial then
$$
\Pr[f=0] = 1 - \left(1 - \frac{1}{|\mathbb{F}|}\right)^d \geq \frac{d}{|\mathbb{F}|} - \frac{d^2}{2|\mathbb{F}|^2}.
$$
(The second bound is Bonferroni's inequality.)
This shows that Schwartz–Zippel is essentially tight when $d$ is small compared to $\mathbb{F}$.
The bound $1-(1-1/|\mathbb{F}|)^d$ is tight for multilinear polynomials, see this answer on math.se.