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.



$X(\mathbb{Z}/p\mathbb{Z})$ versus $\{X(\mathbb{Z})\pmod{p}\}$
I found the MSRI survey *[Strong approximation for algebraic groups](http://library.msri.org/books/Book61/files/70rapi.pdf)* by
Andrei Rapinchuk very well-written and informative (and it has a lengthy bibliography for further exploration).


To answer your question: the cubic hypersurface $X\subset \mathbb{A}^3$ defined by $3x^3 + 4y^3 + 5z^3 =0$ has only one $\mathbb{Z}$-point $(0,0,0)$ but has *more* than one $\mathbb{Z}/p\mathbb{Z}$-points for prime $p$. 


For example, when $p=2$ the locus is $\{(0,0,0),(0,1,0),(1,0,1),(1,1,1)\}$.


So your inequality is generally *strict*.


For a variety $X$ defined over $\mathbb{Z}$, say that $X$ has *strong approximation* if the reduction map $X(\mathbb{Z})\to X(\mathbb{Z}/m\mathbb{Z})$ is surjective for all $m\geq 1$.


Some necessary conditions for this property are that $X$ is absolutely irreducible, and also that the $\mathbb{Z}$-points are Zariski dense. 


An example where strong approximation *does* hold is $\mathrm{SL}\_2$; and more generally simply-connected algebraic groups and homogeneous spaces whose $\mathbb{Z}$-locus is Zariski dense (see the above reference for a precise formulation).


