Let $\phi: \mathbb{A}^n \rightarrow \mathbb{A}^m$ be a polynomial map, where each coordinate is a polynomial defined over $\mathbb{Q}$. Suppose $\phi$ is dominant so the Zarishi closure of $\phi(\mathbb{A}^n)$ is equal to $\mathbb{A}^m$.
I was wondering are there things we can say about the image of rational points $\phi(\mathbb{A}^n(\mathbb{Q}))$? I assume we can not say that $\phi(\mathbb{A}^n(\mathbb{Q})) = \mathbb{A}^m(\mathbb{Q})$ but can we say something about how these points are distributed in $\mathbb{A}^m(\mathbb{Q})$? Thank you.