38 problems

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…

For every polynomial $f\in\mathbb{C}[x_1,x_2,x_3,x_4]$ satisfying $\det\bigl(\operatorname{Hess}(f)\bigr)\in\mathbb{C}^{\times}$, the gradient map…

Let $f\colon\mathbb{R}^2\to\mathbb{R}$ be a Morse polynomial of degree $4$, and consider its principal part. The principal part may be non-discriminant positive definite, vanish on…

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…

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…

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

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 $f:\mathbb{R}^n\to\mathbb{R}^n$ 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 $cA$ 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…

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…

Let $n\geq 1$ and let $A=\{a_{ij}\}_{1\leq i,j\leq n}$ be an $n\times n$ complex matrix. Define the map $T:\mathbb{C}^n\to\mathbb{C}^n$ by … Such a map is called a Drużkowski map.…

Let $f:{\mathbb{C}}^n\rightarrow{\mathbb{C}}^n$ be a polynomial map, meaning that each component $f_i:{\mathbb{C}}^n\rightarrow{\mathbb{C}}$ is polynomial. Jacobi conjecture. If th…

The Valqui–Guccione–Guccione criterion. Then

The Newton-polygon criterion. Then $[F,G]=0$.

Sidon-polynomial classification conjecture. If $P\in\mathbb{F}_p[x]$ is a Sidon polynomial, then either

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 $K$ 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 map. Recall that a real number…

Chamberland–Meisters conjecture. Then $F$ is injective. This conjecture is presented as a sufficient condition for injectivity after the real Jacobian conjecture was disproved; its…

Let $F:\mathbb{R}^n\to\mathbb{R}^n$ be a polynomial map. Real Jacobian conjecture. If … then $F$ is injective. The source explains that this real analogue is false: Pinchuk constru…