Consider a complex polynomial map $f: \mathbb{C}^p \to \mathbb{C}^q$ for some $p \geq q \geq 1$ (not necessarily equal).

**What is a sufficient condition for $f$ to be surjective?**

I am aware of some necessary conditions. For example, one must of course assume that the components $f_1, \dotsc, f_q : \mathbb{C}^p \to \mathbb{C}$ are algebraically independent. (If this is not the case, then $f(\mathbb{C}^p)$ is contained in an affine algebraic variety). When they are indeed independent, one knows that $f(\mathbb{C}^p)$ is dense in $\mathbb{C}^q$ (see When are complex polynomial maps almost surjective?). However, I wonder if conditions are known ensuring that the image is **equal** to $\mathbb{C}^q$. The algebraic independence is not sufficient by itself, as shown by the counter-example $f(z_1, z_2) = (z_1, z_1 z_2)$.