Several Complex Variables Problem List — Problem 2

Definitions and mathematical statement

Let $P=(P_1,P_2):{\mathbb{R}}^2\to{\mathbb{R}}^2$ be a polynomial mapping, with $P_1,P_2\in{\mathbb{R}}[x,y]$. Use the total-degree convention \[ \deg P=\max\{\deg P_1,\deg P_2\},\qquad J_P(x,y)=\det\begin{pmatrix} \partial P_1/\partial x & \partial P_1/\partial y\\ \partial P_2/\partial x & \partial P_2/\partial y \end{pmatrix}. \] The supplied source asks for the lowest degree of a counterexample to the real Jacobian conjecture in dimension two. Its wording is defective: it says that the Jacobian "near vanishes." If this is intended to mean "never vanishes," and if "invertible" means globally bijective, the requested statement is:

Determine the smallest integer $d\geq 1$, provided such an integer exists, for which there is a polynomial mapping $P:{\mathbb{R}}^2\to{\mathbb{R}}^2$ satisfying \[ \deg P=d,\qquad J_P(x,y)\neq 0\quad\text{for every }(x,y)\in{\mathbb{R}}^2, \] while $P$ is not a bijection from ${\mathbb{R}}^2$ onto ${\mathbb{R}}^2$.

This is a conditional reconstruction, not an unqualified transcription. The supplied wording does not resolve the defective Jacobian condition or specify whether failure of invertibility means failure of injectivity, failure of bijectivity, or failure to possess a polynomial inverse. These meanings must be distinguished when confirming the intended minimization problem. The source marks the item "New" and refers to Section 8.2; the contents of that section were not supplied.

Short English statement

Find the least total degree of a polynomial map of the real plane whose Jacobian determinant is everywhere nonzero but which fails the intended global invertibility condition. This reading requires confirmation because the supplied text says "near vanishes" and leaves the meaning of invertibility unspecified. The registry labels the problem open; that status is supplied metadata, not an independently verified assessment.

Sources

• Local UnsolvedMath registry, permanent ID $800206$, source code catalogue entry, title Several Complex Variables Problem List --- Problem 2. The supplied record attributes the problem to Buzzard, Fornaess, and Krantz, with proposed year 1995, and labels it open.
• Buzzard, Fornaess, and Krantz, The Several Complex Variables Problem List (1995), Problem 2, with the source's cross-reference to Section 8.2: https://arxiv.org/abs/math/9509202. This reference and attribution are supplied by the record; the source was not independently consulted for this draft.
• Supplied provenance records access on 2026-10-16 and extraction by arxiv_latex. This is a supplied dated access note, not an independently verified source check or current literature assessment.