Csdrhrt

I am reading Gathmann's online notes.

Here, we define the ring of regular functions on an open neighborhood $U subset X$ by
$$mathcal{O}_{X}(U):= bigcap_{x in U} mathcal{O}_{X,P}$$

where $mathcal{O}_{X,p}:={f/g :f,g in k[X], g(p) neq 0}$.

Note that this does not mean that $f/g$ are globally quotients of polynomials. In particular consider $X=V(x_1x_4-x_2x_3)$ and $U$ is the subset where $x_2 neq 0$ or $x_4 neq 0$. In this case,
$frac{x_1}{x_2} =frac{x_3}{x_4}$ are both regular functions that are given "locally" as quotients of polynomials.

For $U:= mathbb A^2 setminus {0}$, we can write every regular function globally as quotients of polynomials since $X=mathbb A^2$.

I want to know if there is a systematic way of telling when you can write these things as globally regular functions.

My guess is that if we take some $U subset mathbb A^n$ and consider $i:X to mathbb A^n$, then $i^{-1}(U):=V$ is open in $X$. There is an induced map

$$i^*:mathcal{O}_{mathbb A^n}(U) to mathcal{O}_X(V).$$

I guess the induced map is $ psi mapsto psi circ i$. Am I just asking when this map is a surjection?