It is quite general and elementary question.


Is it possible that some holomorphic functions $f\_1,\cdots,f\_m $ on a region $\Omega $ of $\mathbb C$ satisfies:


Whenever $(f\_1(z), \cdots, f\_m (z)) $ is a zero of some polynomial $p \in \mathbb Q [x\_1, \cdots, x\_m]$ for some $z \in \Omega $, then $p (f\_1,\cdots,f\_m)=0$.


Constant functions satisfy this property obviously, so I wonder the existence of non-cobstant maps of certain property.


And what about 'continuous functions', not 'holomorphic'?


I asked the same one at MSE but I didn't get an answer.



Algebraic independence of certain values implies algebraic independence of functions?
EDITED: 


No, it is not possible if $f\_1, \ldots, f\_m$ are not all constant.


Suppose wlog $f\_1$ is not constant. I'll ignore $f\_2, \ldots, f\_m$ and consider polynomials $p(f\_1(z))$. For convenience I'll omit the subscript and
call this $p(f(z))$. $f(\Omega)$ is an open subset of $\mathbb C$, so it contains
$\alpha + \beta i$ for some $\alpha, \beta \in \mathbb Q$, i.e. there is some
$z\_0$ such that $f(z\_0) = \alpha + \beta i$, and so 
$p(f(z\_0)) = 0$ where $p(w) = (w-\alpha)^2 + \beta^2$.


