# Schwartz-Zippel lemma question

Schwartz-Zippel lemma is as follows. Let $$f(x_1,\ldots,x_n)$$ be a polynomial of total degree at most $$d$$ over a field $$\mathbb{F}$$ and assume that $$f$$ is not identically zero. Pick uniformally at random elements $$r_1,\ldots,r_n$$ from the field $$\mathbb{F}$$ then probability that

$$f(r_1,\ldots,r_n) = 0 \le \frac{d}{|\mathbb{F}|}$$

Let us assume that $$f$$ is multiplinear which means each of the monomial in the polynomial $$f$$ is of same degree. My question is as follows. Is the following bound (given below) tight or we can come up with a better bound?

$$f(r_1,\ldots,r_n) = 0 \le \frac{d}{|\mathbb{F}|}$$

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.