I already got a proof for the fact that if a polynomial map is surjective then it is also injective. However, I used the invariant dimension of a ring and I want a simpler proof. Bravo for any try. For preciseness, the statement of the fact is as follows:

Statement: Consider two polynomial rings $k[x_1,...,x_n], k[y_1,...,y_n]$. Let $\Phi: k[x_1,...,x_n] \rightarrow k[y_1,...,y_n]$ be a $k$-algebra homomorphism. If $\Phi$ is surjective then $\Phi$ is also injective.