75 problems

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 $f:\mathbb{R}^n\to\mathbb{R}^n$ 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 $\mathbb K$ be a field, let $n$ be a positive integer, and let $f$ be a multilinear polynomial evaluated on the full matrix algebra $M_n(\mathbb K)$. Kaplansky–L'vov conjecture…

Let $cK$ be either $cC$ or $cR$, and let $cPKknp$ denote the space of degree-$d$ polynomial maps from $cK^n$ to $cK^p$, with topological type defined by composition with homeomorph…

Maubach–Willems conjecture. There are infinitely many finite extensions $K/\mathbf{F}_q$ for which

Let $k$ be a field of characteristic $0$, let $n>1$, and let $r\in\{2,\dots,n\}$. For polynomials $f_1,f_2,\dots,f_r\in k[x_1,\dots,x_n]$, write…

Let $f\colon\mathbb R^n\to\mathbb R^n$ be a polynomial map with nowhere zero Jacobian determinant. Let $S_f$ denote the set of points where $f$ is not proper, and write…

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 $p$.

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 $d\geq 2$ (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…

Zaidenberg's quasihomogeneity conjecture. Up to a polynomial automorphism of $mathbb C^n$, the polynomial $p$ is equivalent to a quasihomogeneous polynomial.

Zaidenberg's isotriviality conjecture. If $X$ is contractible, then $p$ is an isotrivial polynomial: its generic fibres are pairwise isomorphic.

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…

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…

Let $\Bbbk$ be a field of characteristic zero and let $\varphi\in\Bbbk[x_1,\dots,x_n]$. Write … If $\det\operatorname{Hess}\varphi\in\Bbbk^{\times}$, the gradient map has a formal…

Let $q=p^m$ and $Q=p^e$. For integers $n$ and $\ell\geq 1$, let $g_{n,q}^{(\ell)}$ be the multivariate polynomial obtained from $g_{n,q}$ by composing with…

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 $f:\mathbb{R}^n\to\mathbb{R}^n$ 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 $cA$ be the parameter space of maps associated with a triple $cbmA\inccIthth$ of integer polytopes, and let $cA_1$, $cA_2$, and $cA_3$ denote the indicated singularity types. L…