Let $k$ be an algebraically closed field of characteristic $p>0$. Suppose $1< e_i <p$ for $i=1,2, \ldots, n$ are integers ($n \ge 2$). What are the conditions on the $e_i$'s so that the rational function
$$ \omega=\frac{1}{\prod_{j=1}^n (x-P_j)^{e_j}}$$
is a derivative of some rational functions in $k(x)$ (or, equivalently, Res$_{P_j}(\omega)=0$ for all $j$) for some $P_j$'s in $k$ pairwise distinct?
I asked a similar question before in Residues of $\frac{1}{\prod_{i=1}^n (x-P_i)^{e_i}}$. At that time, I conjectured that $\omega$ is an "exact differential form", i.e., $\omega$ is a rational derivative for some $P_j$'s, iff $\sum_{j=1}^n e_j \ge p+n$. The ''$\Rightarrow$'' direction was proved by Fedor Petrov in that thread. However, there are some counter-examples for the ''$\Leftarrow$'' direction due to Gjergji Zaimi in the same thread. The converse still holds for $n=2$ and $n=3$ though (but no longer true for $n=4$ as there is a counter example).
Note that, if the $e_j$'s are given, one can check whether there is an exact differential of the form $\omega$ using Gröbner Bases. In particular, suppose $c_j \in k(P_1, \ldots, P_n)$ is the residue of $\omega$ at $P_j$. Then $\omega$ is an exact differential form if and only if $1$ is not a generator of the reduced Gröbner Bases of $(c_1, \ldots, c_n, 1-\prod_{i \neq j}(P_i -P_j))$.
This question arises from the study of deformations of Artin-Schreier covers ($\mathbb{Z}/p$-covers of the projective line in characteristic $p$). Given $\psi_s$ an Artin-Schreier cover that is branched at exactly one point $b$ with ramification jump $e-1$, and $\{e_1, \ldots, e_n\}$ is a partition of $e$ where $e_i \not \equiv 1 \pmod p$. We show that there exists a deformation $\psi$ over $k[[t]]$ of $\psi_s$, whose generic fiber is branched at $n$ points $Q_1, \ldots, Q_n \in k[[t]]$, which specialize to $b$, have the same $t$-adic valuation, and have ramification jumps $e_1-1, e_2-1, \ldots, e_{n}-1$ if and only if there exists an exact differential of the form
$$ \frac{dt}{\prod_{i=1}^n(x-q_i)^{e_i}}, $$
where $q_i$'s are distinct elements of $k$. Hence, understanding this kind of exact differential forms can help answer questions about deformations and moduli spaces of wildly ramified Galois covers.
