I work with the subset $Z\subset\mathbb C^m$ which is the image of $\mathbb C^n$ under vector-polynomial map $f(x)=(p_1(x),...,p_m(x))$, that is $f$ sends $x\in\mathbb C^n$ to $f(x)\in Z\subset\mathbb C^m$.
Is the following true?
1) $Z$ is IRREDUCIBLE constructible subset of $\mathbb C^m$.
2) the dimension of $Z$ does not exceed $\min(m,n)$
3) the closure of $Z$ is the IRREDUCIBLE variety of dimension not high than $n$?
For 1) I know that the constructibility of $Z$ follows from the Chevalley theorem and the irreducibility follows from the fact that $f$ is a continuous map.
Statement 2) is intuitively clear, but I do not know how to prove it.
For 3) (if it is true) I have no any ideas.