Let $P_1$,...,$P_m$ be polynomials in $n$ variables with coefficients in $\mathbb{Z}$ and consider the set $$X(\mathbb{Z})=\{(x_1,...,x_n)\in \mathbb{Z}^n \ |\ P_i(x_1,...,x_n)=0 ~ ,\ \forall i \in\{1,...,m\}\}.$$

In algebraic geometry, we are often interested in the number of elements in $$X(\mathbb{F}_{p})=\{(x_1,...,x_n)\in \mathbb{F}_p^n \ |\ P_i(x_1,...,x_n)=0 \pmod{p},\ \forall i \in\{1,...,m\}\}.$$

Are there known results about $$\{X(\mathbb{Z})\pmod{p}\}:=\{(x_1,...,x_n) \pmod{p}\ |\ (x_1,...,x_m)\in X(\mathbb{Z})\}~ ?$$

Certainly, $|\{X(\mathbb{Z})\pmod{p}\}|\leq |X(\mathbb{F}_{p})|$. Can we say more?

I'm interested in any notes/papers on the subject.