Let $X $ be a smooth affine subvariety of $(\overline{\mathbb{F}_q})^n$ defined by a prime ideal $I$. Let $f$ $\in \mathbb{F}_q[x_1,\ldots,x_n]$ be a polynomial such that $f \notin I$.

Let $r_1, \ldots, r_m$ be selected independenty and uniformly among all $\mathbb{F}_{q^k}$-rational points of $X$ for some big $k$.

Is it true that $\Pr[f(r_1)=0, \ldots, f(r_m) =0]$ is small for rather big $m$?

It would be a generalization of

Schwartz-Zippel lemma:

Lemma(Schwartz, Zippel).

Let $P\in F[x_1,x_2,\ldots,x_n]$ be a non-zero polynomial of total degree $d \geq 0$ over a field, $F$. Let $S$ be a finite subset of $F$ and let $r_1, r_2, \dots, r_n$ be selected at random independently and uniformly from $S$.

Then $\Pr[P(r_1,r_2,\ldots,r_n)=0]\leq\frac{d}{|S|}.$

UPD: I think I have understood why this must be true: $f \cap V$ is the variety with smaller dimension than $V$ and it has much smaller $\mathbb{F}_{q^k}$-rational points than $V$ (Number of rational points in a non-smooth variety). However I can not get an explicit estimation...