111 problems

Four-variable Hessian conjecture ($\mathrm{HC}_4$). Must $\varphi^L$ be a polynomial? Equivalently, must every four-variable gradient polynomial map with constant nonzero Jacobian…

Let $n$ be a positive integer, and let $F\colon\mathbb C^{n}\to\mathbb C^{n}$ be a polynomial map with constant nonzero Jacobian determinant. Such a map is called a Keller map. The…

Keller's Jacobian conjecture. If $F$ has $\operatorname{Jac} F\in\Bbbk^{\times}$, then $F$ is invertible, with polynomial inverse.

Let $F=(f_1,\ldots,f_n):\mathbb{R}^n\to\mathbb{R}^n$ be a polynomial map, and let $DF(\mathbf{x})$ denote its Jacobian matrix at $\mathbf{x}$. The real Jacobian conjecture. If … fo…

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…

Let $f:\mathbb{R}^n\to\mathbb{R}^n$ be a real polynomial map, and let $f'(x)=\det\operatorname{Jac}_x f$. Strong real Jacobian conjecture. If … for every $x\in\mathbb{R}^n$, then…

Let $\mathcal{R}=\mathbb{C}[x,y]$, and let $N^0(F)$ denote the Newton polygon of $F$ with the origin as reference point. Let $a,b\in\mathbb{Z}_{>0}$ be relatively prime with…

Let $A$ be a commutative binary complex algebra, meaning that its binary operation is symmetric. Generalized Jacobian conjecture for quadratic mappings. If $A$ is Engel, then $A$ i…

Let $A$ be an algebra over a field $\mathbf{k}$ equipped with symmetric $\mathbf{k}$-linear operators $\Psi_\ell$ for $\ell=1,\ldots,m$. A Yagzhev algebra is the algebraic structur…

Let $\Lambda\in\mathbb C[\partial_1,\ldots,\partial_n]$ be a differential operator with constant coefficients, and let $P,Q\in\mathbb C[x]$. Zhao's Generalized Vanishing Conjecture…

Let $n>0$ be an integer. A polynomial map from $\mathbb{C}^n$ to itself is called unramified when its Jacobian function is invertible. The classical Jacobian conjecture. Every unra…

Jacobian conjecture. If $J(P,Q)\equiv c\neq 0$ is a non-zero constant, then $F$ is invertible.

Polar-curve critical-locus conjecture. Under these hypotheses, every polynomial function $g$ has critical locus whose closure in ${\bf X}$ contains $p$.

Nilpotent fixed-point conjecture. A nilpotent mapping $N:\mathbb C^n\to\mathbb C^n$ possesses at most one fixed point.

Retract conjecture. If the corresponding Jacobian matrix of $p$ and $q$ is invertible, then $K[p]$ is a retract of $K[x,y]$.

One-singular-reduction conjecture. If $p\in P_2$ has a unimodular gradient, then one can get

Conjecture “R”. If $p$ has a unimodular gradient, then $K[p]$ is a retract of $K[x,y]$.

Let $n\ge 2$, let $d\ge 1$, and let ${\bf k}$ be a field. A polynomial map $\phi:{\mathbb A}^n_{\bf k}\to{\mathbb A}^n_{\bf k}$ has degree at most $d$ if its coordinate polynomials…

Let $K$ be a unital commutative ${\mathbb Q}$-algebra, let $z=(z_1,\ldots,z_n)$ be commuting variables, and let $H(z)\in K[z]^{\times n}$ be homogeneous of degree $d=3$. Let…

Let $K$ be a unital commutative ${\mathbb Q}$-algebra, let $z=(z_1,\ldots,z_n)$ be commuting variables, and let $H(z)\in K[z]^{\times n}$ be homogeneous of degree $d\geq 2$ (equiva…

Let $P:\mathbb C^2\rightarrow\mathbb C^2$ satisfy $J(P)\equiv 1$. Suppose there exists an algebraic function defined on some open set ${\cal U}$ such that $P\circ Q=P$, and suppose…

Homogeneous vanishing conjecture. Both of the following vanishings should hold: 1. $\Delta^mP^{m+1}=0$ for every $m>\alpha_{[n,d]}$. 2. For every $k\geq1$, $\Delta^mP^{m+k}=0$ for…

Let $P(z)\in {\mathbb C}[[z]]$ be a homogeneous Hessian-nilpotent polynomial of degree $d\geq 2$, and let $Q_t(z)$ denote its deformed inversion pair. Vanishing conjecture. The uni…

Polynomiality conjecture. The solution $U_t(z)$ must be a polynomial in both $z$ and $t$. This conjecture links the nilpotent-Hessian condition to polynomial solutions of the invis…

Cubic nilpotent-Jacobian conjecture. If $JH(z)$ is nilpotent, then $N_t(z)$ is a polynomial solution in both $t$ and $z$. This is presented as an equivalent formulation of the Jaco…