67 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 $z=(z_1,z_2,\dots,z_n)$, let $D_i=\frac{\partial}{\partial z_i}$, and let $\Delta_n=\sum_{i=1}^n D_i^2$ be the Laplace operator on the polynomial algebra…

Let $z=(z_1,z_2,\dots,z_n)$ be free commutative variables, let $D_i=\frac{\partial}{\partial z_i}$, and let … be the Laplace operator on $\mathbb C[z]$. A polynomial $P(z)$ is Hess…

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 $P:\mathbb C^2\rightarrow\mathbb C^2$ 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…

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…

Let $r(x)$ be a monic polynomial of degree $N+1>1$ with distinct roots, and let $\lambda(x)$ and $\mu(x)$ be the unique polynomials satisfying … with $\deg\mu(x)\leq N-1$ and…

Divergence criterion. The map $F(z)$ is polynomial if and only if $G(z)$ is polynomial. The source presents this as an algebraic formulation equivalent to the Jacobian conjecture,…

Let $K$ be a field of characteristic $0$, let $\varphi\in K[x_1,\ldots,x_n]$ be a polynomial whose Hessian determinant is a nonzero constant, and let $\bar\varphi$ be its formal Le…

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

Adjamagbo's separable Jacobian conjecture. Assume that

Let $A=\bigoplus_{n\in {\mathbb{N}}}S^{dn}{\mathbb{C}}^N\otimes S^k({\mathbb{C}}^N)^*$ be a graded $SL(N,{\mathbb{C}})$-algebra. Let $P^+$ denote the set of dominant highest weight…

Let $n$ be a positive integer, and let $f:{\mathbb{C}}^n\rightarrow {\mathbb{C}}^n$ be a polynomial map with Jacobian $J(f)=1$. Suppose that, for each $i$, $f_i=x_i-h_i$, where…

Let $n$ be a positive integer, let $k$ be an algebraically closed field of characteristic zero, and let $f:k^n\rightarrow k^n$ be a polynomial map, meaning that each component…

Let $K$ be a compact connected Lie group, and let $f$ and $h$ be complex-valued finite-type functions on $K$. Let $dk$ denote Haar measure on $K$. Mathieu's conjecture. If … for al…

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…

Kernel conjecture.

Let $p$ be a prime, let $n\geq 1$, and let … satisfy $\det JF\in\mathbb{Z}_p^\times$. A vector in $\mathbb{Z}_p^n$ is unimodular when the ideal generated by its entries is…

Let $f:\mathbb{R}^n\to\mathbb{R}^n$ be a real polynomial map with Jacobian determinant $f'(x)>0$ for every $x\in\mathbb{R}^n$, and let $S_f$ denote its non-properness set. Real top…

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 $\mathbb{F}$ be either $\mathbb{R}$ or an algebraically closed field of characteristic $0$, and consider the morphism $\phi$ of the affine superspace $\mathbb{A}^{m|n}$ over…

Let $\overline{\mathcal{R}}=\mathbb{C}[x^{1/\mathbf d},y]$ for a fixed positive integer $\mathbf d$, and let $\overline{T}_{m,n,a}$ be the class defined in the paper. Suppose…

Let $Z$ be the innermost polynomial and $\zeta(F)$ the inner polynomial arising from the paper's construction. For a convex polygon, the NE vertex is the uppermost vertex with larg…