A characterization of differential operators in the ring of complex polynomials
Włodzimierz Fechner, Eszter Gselmann
TL;DR
The paper addresses the problem of characterizing all Leibniz-rule operators on the polynomial ring $\mathscr{P}(\mathbb{C})$ without assuming linearity. It develops a localization-based analysis and proves a complete characterization: such an operator $T$ is governed by sequences $(\psi_k)_{k\ge0}$ and $(\tilde{\varphi}_k)_{k\ge0}$, with each $\tilde{\varphi}_k$ satisfying $\tilde{\varphi}_k(ab)=a\tilde{\varphi}_k(b)+b\tilde{\varphi}_k(a)$, yielding an explicit root-based expansion for $T$ when $p(z)=a\prod_{j=1}^N(z-z_j)$. The paper further derives localization-like consequences and provides corollaries on degree behavior, notably that a degree-decreasing $T$ forces a form proportional to the derivative on monomials and, more generally, not-increasing-degree cases give $T(p)=c_p p'+d_p p$. Additionally, it presents explicit families of Leibniz-rule maps (e.g., $N$, $F$, $P$, $E$, $K$, $Q$) to illustrate the breadth of possible operators beyond differential operators.
Abstract
The paper aims to provide a full characterization of all operators $T\colon \mathscr{P}(\mathbb{C}) \to \mathscr{P}(\mathbb{C})$ acting on the space of all complex polynomials that satisfy the Leibniz rule \[ T(f\cdot g)= T(f)\cdot g+f\cdot T(g) \] for all $f, g\in \mathscr{P}(\mathbb{C})$. We do not assume the linearity of $T$. As we will see, contrary to the well-known theorems for function spaces there are many other solutions here, not only differential operators. From our main result, we also derive two corollaries, showing that in some special cases operators that satisfy the Leibniz rule have some particular form.