Let $K$ be a field and $\varphi: K[X_1,X_2,...,X_n] \to K[Y_1,Y_2,...,Y_m]$ a polynomial ring morphism. Assume $n, m \ge 2$. By definition $\varphi$ endows $K[Y_1,Y_2,...,Y_m]$ with a $K[X_1,X_2,...,X_n]$-module structure.
Are there some available criterions to decide when the $K[X_1,X_2,...,X_n]$-module $K[Y_1,Y_2,...,Y_m]$ induced in this way by $\varphi$ a flat $K[X_1,X_2,...,X_n]$-module?
I want also to note that this generalizes this question: Polynomial map $K[t] \to K[t]$ induces flat module structure
as far as we consider the case with more then one undeterminants. The case $n=m=1$ had always a positive answer since $K[X]$ is a PID and in this setting flat = torsion-free. For $n \ge 2$ $K[X_1,X_2,...,X_n]$ is not a PID, so the criterion is not appliable.
Another approach is to use a lemma that states that if $R \to R', S \to S'$ are flat modules, then $R \otimes S \to R' \otimes S'$ is a flat $R \otimes S$-module. The point is that I think that not every polynomial map $\varphi: K[X_1,X_2,...,X_n] \to K[Y_1,Y_2,...,Y_m]$ arises from such atomar pieces $\varphi_i: K[X_i] \to K[Y_i]$. Therefore, this approach also fails.