There are several notions of rank/dimension defined on differential fields. However, we do not have a reasonable way to estimate these typically ordinal valued invariants. Especially, we do now know a lower bound for the Lascar Rank, an invariant coming from model theory. It turned out that the question of finding a lower bound is related to the following question.
Let $K\subseteq K\langle \eta\rangle$ be a partial differential field extension of characteristic zero. Suppose that the Kolchin polynomial $\omega_{\eta/K}(t)$ is of degree $n>0$. Is it true that for any $k < n$ there is a $\nu$ in $K\langle \eta\rangle$ such that the degree of $\omega_{\nu/K}(t)$ is $k$? (Here $\eta$ and $\nu$ are finite tuples).
Now, as differential algebra is not a very popular subject and most people do not know what a Kolchin polynomial is, I will also ask a very similar question in commutative algebra.
Let $S$ be a graded commutative algebra over $K[X_1,\ldots,X_d]$ where $K$ is a field of characteristic zero and let $H_S(t)$ be its Hilbert polynomial. Suppose that $\deg (H_S(t))=n>0$. Is it true that for any $k< n$ there is a graded subalgebra $T$ of $S$ such that $\deg(H_T(t))=k$?
