A basis conjecture for fixed points of the Cartier operator

Let $R\in\mathbb{F}_q[z]$ have degree at least $1$, and write

$$R=cR_1^{e_1}\cdots R_k^{e_k}$$

for its factorization into irreducibles. For a polynomial $S=\sum_{j=0}^s c_jz^j$ with $c_s\neq0$, define

$$\Delta(S)=\sum_{j=0}^s(s-j)c_jz^j.$$

The operator $\lambda_0$ is defined by $\lambda_0(S)=\Lambda_0(SR^{q-1})$.

The fixed-point basis conjecture. For every $i\in\{1,2,\dots,k\}$, the polynomial

$$R_1^{e_1}\cdots R_{i-1}^{e_{i-1}}\Delta(R_i^{e_i})R_{i+1}^{e_{i+1}}\cdots R_k^{e_k}$$

is a fixed point of $\lambda_0$. Moreover, if $R$ is not divisible by $z$ and $e_i\not\equiv0\pmod p$ for every $i$, where $p$ is the characteristic of $\mathbb{F}_q$, these $k$ fixed points together with $R$ are linearly independent.

This conjecture proposes explicit fixed points and, under the stated hypotheses, a linearly independent family of $k+1$ elements. It is intended to provide a basis for the fixed-point space in the setting described in the paper; its status is not resolved in the source.