33 problems

Nowicki's conjecture. The algebra $K[X_n,Y_n]^\delta$ is generated by $x_1,\ldots,x_n$ and

Flenner–Zaidenberg rigidity conjecture. The affine Pham–Brieskorn hypersurface $X_{a_0,\dots,a_n}$ is rigid if and only if $\min\{a_0,\dots,a_n\}\geq 2$ and at most one element $i$…

Exponential Generators Conjecture. The group $\mathrm{GA}_n(K)$ is generated by the affine automorphisms together with the exponentials $\exp(D)$ of locally nilpotent derivations.

Makar-Limanov invariant stability conjecture.

Liendo's FML rationality conjecture. If

Pham–Brieskorn rigidity conjecture. The coordinate ring $\mathcal O_{\bar Y}(\bar Y)$ is rigid if and only if $a_1\geq 3$.

Rigidity conjecture. It is conjectured that, for every $n\ge 3$ and every $(a_1,\dots,a_n)\in\Gamma_n$, the ring $B_{\mathbb{C};a_1,\dots,a_n}$ is rigid.

Exponential Generators Conjecture. The group $\operatorname{Aut}_R R[X]$ is generated by $\operatorname{Aff}(R,n)$ and the exponential automorphisms of $R[X]$.

Let $X$ be an affine variety of dimension at least $2$, and let $D_1,\ldots,D_k$ be locally nilpotent derivations on $\mathbb K[X]$. For each $i$, let $H_i=\exp(tD_i)$ be the corre…

Let $K$ be a field of characteristic zero and let $K[T]=K[t_1,\dots,t_n]$. A derivation $D$ of a ring is locally nilpotent if for every $a$ there is an integer $m(a)$ such that…

Let $Y$ be a variety, let ${\mathbb K}[Y]$ be its coordinate ring, let $I$ be the ideal generated by the images of all locally nilpotent derivations on $Y$, and let…

Let $Y$ be a variety, let ${\mathbb K}[Y]$ be its coordinate ring, and let $I$ be the ideal of ${\mathbb K}[Y]$ generated by the images of all locally nilpotent derivations on $Y$.…

Freudenburg's conjecture. Every image ideal $I_n$ is principal.

Conjugacy-to-triangularity conjecture. For every $\phi\in\mathcal{E}_3(k)\cap\operatorname{T}_3(k)$, there exists $\sigma\in\operatorname{T}_3(k)$ such that

Let $X$ be a horospherical complexity-zero variety, let $\tau$ be a face of its cone $\sigma^\vee$, and let $O_\tau$ be the corresponding $G$-orbit. Let $p_1,\ldots,p_k$ be the pri…

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

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

Freudenburg's conjugacy conjecture. Every maximal Lie subalgebra $\mathcal{A}\subset\operatorname{LND}(\mathbb{K}[x_1,\ldots,x_n])$ is conjugate to $\mathfrak{T}$. This is the uniq…

Let $X$ be an affine variety and let $A=\mathcal{O}_X(X)$ be its structure algebra. For locally nilpotent derivations $\partial_1,\ldots,\partial_k\in\operatorname{LND}(A)$, set ……

Elementary derivation conjecture. All $\deg$-homogeneous locally nilpotent derivations of a trinomial algebra are elementary.

Let $B=k^{[3]}$. Given a nonzero locally nilpotent derivation $D\in\operatorname{LND}(B)$, set $A=\ker D$. Miyanishi's theorem asserts that $A\cong k^{[2]}$. Freeness conjecture. T…

Commuting Derivations Conjecture. Then the common kernel of the derivations in $D$ is generated by a coordinate of $k[x_1,\ldots,x_n]$.

Slice-coordinate conjecture. Such a slice $s$ must automatically be a coordinate.

Rank-one derivation conjecture. There always exists $D\in\operatorname{LND}_k k[\bf x]$ with $\operatorname{rank}D=1$ such that $D(f)=0$ and $D$ is tamely triangularizable.

Rank-three wildness conjecture. If $\operatorname{rank}D=3$, then $\exp D$ is wild.