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](https://en.wikipedia.org/wiki/Schwartz%E2%80%93Zippel_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](https://mathoverflow.net/questions/237145/number-of-rational-points-in-a-non-smooth-variety)). However I can not get an explicit estimation... 



Schwartz-Zippel lemma for an algebraic variety
Let $d=\dim(X)$ and let $Y=X\cap\{f=0\}$, so $\dim(Y)=d-1$. Then the Lang-Weil estimate says that $\#X(\mathbb F\_{q^k})=q^{dk} + O(q^{k(d-1/2)})$, and applying it to each irreducible component of $Y$ gives $\#Y(\mathbb F\_{q^k})=O(q^{(d-1)k})$. Your probability is the ratio $\#Y(\mathbb F\_{q^k})^m/\#X(\mathbb F\_{q^k})^m$, if I understand correctly, which equals $O(C^mq^{-mk})$, which is indeed small when $m$ is large, provided you choose $k$ large enough. If you want to include an error term, which would of course depend on the ideal defining $X$ and the function $f$ defining $Y$, then the big-$O$ constant may be taken to be the number of components $\nu(Y)$ of $Y$, i.e., something like $\text{Prob}\le \nu(Y)^mq^{-mk}+O(q^{-(m+1/2)k})$


