TITLE: Number of solutions to polynomial congruences
QUESTION [16 upvotes]: Suppose I have $R$ homogeneous polynomials $F_1, ..., F_R$ with integer coefficients. Let $V$ be the affine variety defined by these polynomials over $\mathbb{C}$. I was wondering if some bound that looks like the following was known?
$$
\#\{ \mathbf{x} \in (\mathbb{Z}/q \mathbb{Z})^n: F_{i}(\mathbf{x}) \equiv 0 (\text{mod }q) \text{  for each } 1 \leq i \leq R \} \leq C q^{\dim V + \delta}  
$$ 
for some $\delta > 0$. Here $C>0$ is independent of $q$. Of course if $\delta = n - dim V$ then it's trivial so I was hoping for something smaller. I would greatly appreciate any comments on this. Thank you very much.

REPLY [3 votes]: The case of $q$ a prime being answered by Igor, let us assume that $q=p^k$ is a power of a prime $p$. In this case there is the following result of Serre, which gives a very partial answer:
Let $d_p$ be the dimension of the $p$-adic analytic space $V$ defined in $\mathbb Z_p^R$ by the equations $F_1=\dots=F_r=0$. Then
$$
\#\{ \mathbf{x} \in (\mathbb{Z}/q \mathbb{Z})^n: F_{i}(\mathbf{x}) \equiv 0 (\text{mod }q) \text{  for each } 1 \leq i \leq R \} \leq C_p q^{d_p}  
$$ 
where $C_p$ is some constant depending on $p$ but not on $k$.
The result is Serre's Théorème 8 in his IHES paper on Chebotarev.
Comparing to what you ask, there are two differences: the exponent is $d_p$ instead of $\dim_{\mathbb C} V$ (but they are the same for almost all $p$, aren't they?). And the constant $C$ depends on $p$, though it is the same for all powers $q$ of a given $p$.