Ax-Grothendieck Theorem states that if $\mathbf K$ is an algebraically closed field, then any injective polynomial map $P:\mathbf K^n\longrightarrow \mathbf K^n$ is bijective.

Question 1. What does the inverse map of $P$ look like ? What kind of map is that ?

$P^{-1}$ need not be polynomial, as the example $x^p$ in $\mathbf F_p^{alg}$ shows.

Question 2. Are there conditions under which $P^{-1}$ is polynomial ?