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. 



Number of solutions to polynomial congruences
First note that you can't do better than the trivial bound $q^n$ in general; for example if $q$ divides all the coefficients of the $F\_i$.


However this silly problem only occurs for finitely many primes (by the Chinese remainder theorem we can reduce to the case of prime powers). So let $p$ be a prime such that $V\_{\mathbb{F}\_p}$ has dimension $d:=\dim V$ (here I denote by $V\_{\mathbb{F}\_p}$ the reduction of $V$ modulo $p$). By Noether normalisation there exists a finite map
$$f: V\_{\mathbb{F}\_p} \to \mathbb{A}^d\_{\mathbb{F}\_p}.$$
Let $C$ be the degree of $f$. Then we have
$$\#V(\mathbb{F}\_p) \leq C\#\mathbb{A}(\mathbb{F}\_p) = Cp^{d}.$$
This is much easier and weaker than the Lang-Weil estimates, which are much deeper and require the Riemann hypothesis for curves.


To treat prime powers, there are some subtleties. One needs to understand the cardinalities of the fibres of the map
$$\#V(\mathbb{Z}/p^k\mathbb{Z}) \to \#V(\mathbb{F}\_p) \quad (\*).$$
If $V\_{\mathbb{F}\_p}$ is smooth, then a version of Hensel's lemma states that the fibres of the map $(\*)$ all have cardinality exactly $p^{d(k-1)}$. This gives the upper bound
$$\#V(\mathbb{Z}/p^k\mathbb{Z}) \leq Cp^{kd}.$$
By "spreading out arguments" (essentially applying Noether Normalistion to the variety over $\mathbb{Q}$ then spreading out the map), one finds an absolute constant $C$ such that
$$
\#\{ \mathbf{x} \in (\mathbb{Z}/q \mathbb{Z})^n: F\_{i}(\mathbf{x}) \equiv 0 (\text{mod }q) \text{ for each } 1 \leq i \leq R \} \leq C^{\omega(q)} q^{\dim V} 
$$ 
holds for those $q$ which satisfy the above conditions on the dimension of $V \bmod q$ and smooth reduction. Here $\omega(q)$ denotes the number of prime divisors of $q$.


