56 problems

Does every polynomial $\phi\in\mathbb{R}[x_1,x_2,x_3,x_4]$ satisfying $\det\bigl(\operatorname{Hess}\phi(x)\bigr)=-1$ and having Lorentzian Hessian, namely…

Let $K$ be a field of characteristic $0$, let $K[x,y,z]$ be the commutative polynomial algebra, and let $K\langle x,y,z\rangle$ be the free associative algebra. An automorphism is…

Let $f$ be the polynomial automorphism under consideration, and let an $n$-pod be a configuration formed by $n$ conjugates of $f$ in the Bass–Serre tree, with consecutive branches…

Adjamagbo's separable Jacobian conjecture. Assume that

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

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…

Rips's conjecture. Linear automorphisms, together with $\psi$, generate the entire tame automorphism group of $K[x_1,\ldots,x_n]$. The characteristic-zero analogue is known, while…

Let $F$ and $G$ be algebraically independent polynomials in $\mathbb{C}[x_1,\ldots,x_n]$. Assume that each generates its own centralizer in $\mathbb{C}[x_1,\ldots,x_n]$, that…

Let $K$ be a field of characteristic $0$, let $d$ be a positive integer, and let $f\colon \mathbb{A}^d_K\to\mathbb{A}^d_K$ be a polynomial automorphism. Blanc–Van Santen's conjectu…

Let $f_0$ and $f_1$ be polynomial automorphisms of ${\mathbb{C}^2}$ with non-trivial dynamics. Write $J_i=J(f_i)$ and $J_i^\star=J^\star(f_i)$ for their Julia sets and assume that…

Let $\operatorname{Aut}(\mathbb C^2)$ denote the space of polynomial automorphisms of $\mathbb C^2$. A dynamical system has a wandering Fatou component when it displays a Fatou com…

Polydegree Conjecture. Then

Let $K$ be a field of characteristic zero and let $K[x]=K[x_1,\ldots,x_n]$. Consider the derivation … Let $\operatorname{Aut}(K[x])_D$ denote the subgroup of polynomial automorphis…

The converse conjecture. The derivation $D$ is simple if and only if

The trivial-centralizer conjecture. If $D$ is a simple Shamsuddin derivation of $K[x]$, then

The finiteness conjecture. If $D$ is a simple derivation of $K[x_1,x_2]$, then

Let ${\rm SA}_n(\mathbb{K})$ denote the group of special polynomial automorphisms, and let ${\rm SLIN}_n(\mathbb{K})$ denote the corresponding subgroup satisfying the special linea…

Let $\mathbb{K}$ be a field, and let ${\rm GA}_n(\mathbb{K})$ denote the group of polynomial automorphisms of affine $n$-space over $\mathbb{K}$. Let ${\rm GLIN}_n(\mathbb{K})$ be…

Let $F:\mathbb{C}[x_1,\ldots,x_n] \to \mathbb{C}[x_1,\ldots,x_n]$ be a $\mathbb{C}$-algebra endomorphism with invertible Jacobian. Write $F_i=F(x_i)$, and let $l_i$ be the degree o…

Let $H$ be an irreducible affine hypersurface in $\mathbb{A}^s\times\mathbb{A}^s$. It is $(s,t)$-grid-free when the associated bipartite algebraic graph contains no complete bipart…

Let $F:K^d\rightarrow K^d$ be a polynomial map with $F_i=X_i+H_i$, where … and … for $1\leq i\leq d$. Write $H=(H_1,\dots,H_d)$ and let $JH$ be its Jacobian matrix. Define polynomi…

Let $k$ be a field of characteristic $p$. A strong Keller map is a polynomial endomorphism in $\operatorname{SKE}_n(k)$ satisfying the universal Keller equations. Composition conje…