Comparison of powers of differential polynomials

Necessary and sufficient conditions are obtained for a polynomial $P$ to be more powerful then a polynomial $Q$. These conditions are formulated in terms of the orders of generalized-homogeneous sub-polynomials, corresponding to these polynomials, and the multiplicity of their zeros. Applying these results, conditions are obtained, under which a monomial $\xi^v$ for a certain set of multi-indices $v\in\mathfrak{R}^*$ can be estimated via terms of a given degenerate polynomial $P$.