4 problems

Zhao's Image Conjecture. If $f\in\mathbb C[\zeta,x]$ satisfies $f^m\in\operatorname{Im}D$ for every positive integer $m$, then for each $g\in\mathbb C[\zeta,x]$,

One-dimensional Image conjecture. The image $\operatorname{Im}D$ is a Mathieu subspace of $A[t]$. The source identifies this as the one-dimensional case of the Image Conjecture and…

Let ${\mathcal M}$ be the subspace of polynomials $f(\xi,z)\in{\mathbb C}[\xi,z]$ such that ${\mathcal Z}(f)$ has no holomorphic part, equivalently such that the Laplace transforma…

The Image Conjecture. For any $f,g\in {\mathcal A}[z]$ such that $f^m\in\operatorname{Im}{\mathcal C}$ for every $m\geq 1$, one has $f^m g\in\operatorname{Im}{\mathcal C}$ for all…