Several Complex Variables Problem List — Problem 55

Definitions and mathematical statement

Let $P=(P_1,P_2):{\mathbb{R}}^2\to{\mathbb{R}}^2$, where $P_1,P_2\in{\mathbb{R}}[x,y]$. Its real Jacobian determinant is \[ J_P(x,y)=\det\begin{pmatrix} \partial P_1/\partial x & \partial P_1/\partial y\\ \partial P_2/\partial x & \partial P_2/\partial y \end{pmatrix}. \] Under the usual total-degree convention, set \[ \deg P=\max\{\deg P_1,\deg P_2\}. \] If "invertible" means invertible as a map from ${\mathbb{R}}^2$ onto ${\mathbb{R}}^2$, define \[ \mathcal D=\left\{\deg P:\begin{array}{l} P\in{\mathbb{R}}[x,y]^2,\\ J_P(a)\ne0\text{ for every }a\in{\mathbb{R}}^2,\\ P:{\mathbb{R}}^2\to{\mathbb{R}}^2\text{ is not bijective} \end{array}\right\}. \] The question, under these conventions, is to determine $\min\mathcal D$, provided $\mathcal D\ne\varnothing$: what is the lowest degree of a real polynomial map of the plane whose Jacobian determinant never vanishes but which fails to be globally invertible?

The supplied statement does not explicitly define the degree convention or whether invertibility requires a polynomial inverse. Thus the displayed formulation is conditional on total degree and global invertibility as a function; requiring a polynomial inverse would give a different formulation. The hypothesis is nowhere vanishing of the real Jacobian determinant, not a requirement that it be constant. The source labels this a new problem in its 1995 list; the supplied registry labels it open, without supplying an independently verified current status.

Short English statement

Determine the smallest degree of a polynomial map ${\mathbb{R}}^2\to{\mathbb{R}}^2$ with a nowhere-zero real Jacobian determinant that is not globally invertible. The displayed formulation uses total degree and interprets invertibility as bijectivity; the source does not explicitly settle these conventions.

Sources

• Local UnsolvedMath registry, permanent ID $800256$, exact source problem code catalogue entry, title "Several Complex Variables Problem List --- Problem 55." The supplied record attributes the problem to Buzzard, Fornaess, and Krantz, with proposed year 1995, and labels its status open.
• Buzzard, Fornaess, Krantz, The Several Complex Variables Problem List (1995), Problem 55, as identified in the supplied provenance: https://arxiv.org/abs/math/9509202. The supplied access date is 2026-10-16 and the extraction method is arxiv_latex. This reference and access information are supplied provenance; no independent source consultation or current literature verification is claimed.