26 problems

The LFED Conjecture. Images of locally finite $K$-derivations and $K$-$\mathcal{E}$-derivations of $K$-algebras are Mathieu subspaces.

Let $K$ be a field of characteristic zero, ${\mathcal A}$ a commutative $K$-algebra, and let $\delta$ be a locally nilpotent (LN) $K$-derivation or a locally nilpotent $K$-…

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 $g\in\mathbb C[\zeta,x]$,

Let $A={\mathbb C}[\xi,z]$, where $\xi=(\xi_1,\ldots,\xi_m)$ and $z=(z_1,\ldots,z_m)$ are two sets of commuting indeterminates. For each $i$, define…

Image conjecture for classical orthogonal polynomials. The subspace $\operatorname{Im}'\Lambda$ is a Mathieu subspace of $\mathbb C[t]$. The source cites special results and relate…

Let $n\geq 1$, and consider the polynomial algebra in $n$ variables over $\mathbb C$. For suitable open subsets of $\mathbb R^n$ and positive measures, the subspace of polynomials…

Let $B\subseteq\mathbb R$ be open, and let $\sigma$ be a positive measure such that $\int_B g(t)\,d\sigma$ is finite for every $g(t)\in\mathbb C[t]$. Define … A subspace $V$ of an…

Let $K$ be a field of characteristic zero, and let $z=(z_1,z_2,\dots,z_n)$ be $n$ commuting indeterminates. For a commutative $K$-algebra ${\mathcal A}$ and a $K$-subspace $V$ of…

Let $K$ be a field of characteristic zero, ${\mathcal A}$ a commutative $K$-algebra, and let $\delta$ be a locally finite $K$-derivation or $K$-${\mathcal E}$-derivation of…

LF derivation image conjecture. The image of every locally finite $K$-derivation or locally finite $K$-${\mathcal E}$-derivation of ${\mathcal A}$ is a Mathieu subspace.

Let $K$ be a field of characteristic zero, not necessarily algebraically closed, and let ${\mathcal A}$ be a $K$-algebra with a decomposition … A decomposition is an additive algeb…

Let $K$ be a field of characteristic zero and ${\mathcal A}$ a $K$-algebra, and let $\delta$ be a $K$-derivation or $K$-${\mathcal E}$-derivation of ${\mathcal A}$. Write…

Strong simplicity conjecture. The only strongly simple algebras are fields over themselves.

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 $\Lambda$ be one of the differential operators associated with the classical orthogonal-polynomial weights described above, and set … Assume that…

Let $B\subset\mathbb R$ be open, let $\sigma$ be a positive measure with finite moments of all orders, and define … Assume that $\sigma$ is not an atomic measure supported at finit…

Integral conjecture. The subspace $V_B(\sigma)$ is a Mathieu subspace of $\mathbb C[t]$. This is one of the one-dimensional Integral and Image Conjectures; the source records speci…

Image Conjecture. The image of $\mathcal D$ is a Mathieu subspace of $B$.

Let $k$ be a field, let $A$ be a commutative $k$-algebra, and let $A[z]=A[z_1,\ldots,z_n]$. Let $D$ be a commuting set of differential operators of the form described in the source…

Let $k$ be a field, let $A$ be a commutative $k$-algebra, and let $A[z]=A[z_1,\ldots,z_n]$. Let $a_1,\ldots,a_n\in A$, and define … A sequence $(a_1,\ldots,a_n)$ is regular when ea…

For $t\in\mathbb C$, let $\mathcal B_t[\xi,z]$ be the algebra on the vector space $\mathbb C[\xi,z]$ with product $\ast_t$, and set … A subspace of a commutative algebra is a Mathi…

Let $t\in\mathbb C$, and let … be the image of the commuting differential operators $(\xi_i-t\partial_i)$. A subspace $\mathcal M$ of a commutative algebra $\mathcal A$ is a Mathie…

Let $B\subset\mathbb R^n$, let $w(z)$ be a weight function, and let $\{u_\alpha\mid\alpha\in\mathbb N^n\}$ be a sequence of orthogonal polynomials over $B$. Let $\mathcal M$ be the…

The generalized Mathieu conjecture. The subspace $\mathcal M_B(\sigma)$ is a Mathieu subspace of $\mathbb C[z]$. The source presents this as a variation of Mathieu's conjecture and…

The generalized image conjecture. The subspace $\operatorname{Im}'\Lambda$ is a Mathieu subspace of $\mathbb C[z]$. The paper notes that this generalizes the image conjecture and p…