I was wondering if this is this is true or not:
If $D$ is a derivation and $x_1,x_2,...,x_n$ are algebraically independent, and $p(x_1,...,x_n)$ is a homogeneous polynomial with all of its monomials being relatively prime (i.e. there is no common divisor between any two of them). Is it true that the minimum $i$ such that $D^i p = 0$ is the minimum $i$ such that ALL of those monomials are $0$?
If yes, how can this be proved? I hope my question is clear.