87 problems

Let $k$ be a field of characteristic zero and let $\operatorname{Aut}_k k[x_1,x_2,x_3]$ be the automorphism group of the polynomial ring in three variables. Define…

Let $\mathcal{R}=\mathbb{C}[x,y]$ and, for $f,g\in\mathcal{R}$, write … A pair $(f,g)$ is a Jacobian pair when $[f,g]\in\mathbb{C}\setminus\{0\}$. Consider the endomorphism…

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.

Automorphism conjecture. Every automorphism of the Lie algebra $W_{\infty}(\mathbb{F})$ is a conjugation by an automorphism…

Let $k$ be a field, let $x$ be an indeterminate, and let $f\in k[x]^{[n]}$ be a residual coordinate, meaning that it becomes a coordinate modulo $x-c$ for each $c\in k$. Dolgachev–…

Let $\mathbf{k}$ be an algebraically closed field, and let $\operatorname{Aut}_{\mathbf{k}}(\mathbb{A}^n)$ denote the group of automorphisms of affine $n$-space over $\mathbf{k}$.…

Let $\Phi:\mathbb{C}^n\to\mathbb{C}^n$ be a polynomial automorphism of order two. The Linearization Conjecture states that there exists an automorphism…

Linearizable generation conjecture. The group generated by the linearizable automorphisms may equal the full automorphism group:

Let $\mathbb{K}$ be a field of characteristic $0$. Consider the polynomial algebra $\mathbb{K}[x_1,x_2,x_3]$; an elementary automorphism is one of the form … where…

Let $K$ be the ground field, let $V(p)$ be a hypersurface in affine $n$-space, and call two subvarieties equivalent if an automorphism of the ambient affine space takes one onto th…

The paper considers automorphisms of polynomial algebras over a field; linear automorphisms are induced by invertible linear changes of variables, and triangular automorphisms are…

Residual-variable conjecture. If $p$ is a residual $x_1$-variable, then $p$ is an $x_1$-variable, meaning

Let $k$ be an algebraically closed field of characteristic zero, and let $H:k^2\to k^2$ be a polynomial mapping whose Jacobian $\operatorname{Jac} H$ is a non-zero constant. Keller…

Adjamagbo's separable Jacobian conjecture. Assume that

Pascal Finite Generators Conjecture. Every element of $\operatorname{Aut}_{su}(K,n)$ belongs to $\langle PF(K,n)\rangle_{\circ}$.

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

Baltazar's finiteness conjecture. The isotropy group of $d$ is finite.

Let $R$ be a commutative ring, let $R[\boldsymbol{x}]=R[x_1,\ldots,x_n]$, and let $\sigma\in\mathop{\rm Aut}\nolimits_R R[\boldsymbol{x}]$. For $r\geq 0$, write…

Let $f$colon ${\mathbb C}^2\to{\mathbb C}^2$ be a dissipative and hyperbolic automorphism, meaning that $|\operatorname{Jac} f|<1$, and suppose that $f$ has no attracting points. L…

Principal-plinth-ideal conjecture.

Invariant-coordinate conjecture. For every such $\phi$, there exists $\sigma\in\operatorname{T}_3(k)$ such that

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

Stable Tameness Conjecture. Every $\phi\in\operatorname{Aut}_k k[\boldsymbol{x}]$ is stably tame. This is a central stabilization question for polynomial automorphisms; the source…

Let $f^{\circ}: \mathbb{A}^N \to \mathbb{A}^N$ be a polynomial automorphism of degree at least two. An $f^{\circ}$-periodic point is isolated if it is isolated in the corresponding…

Let $f\colon\mathbb{R}^{+}\times\mathbb{R}^{n}\to\mathbb{R}^{n}$ be as in the system $\dot{x}=f(t,x)$, and assume that it satisfies (G1) and (G3$^{*}$). Here (G3$^{*}$) requires th…