Conjecture Fieldbook

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Two-dimensional Jacobian conjecture

Canonical problem dossier

For polynomial maps in two variables over characteristic zero, does a constant non-zero Jacobian determinant force a polynomial inverse?

Status
Open
Field
Algebra
Verification
Reviewed dossier
Source grade
secondary
Reviewed
2026-07-27T18:04:33.834Z
Reviewer
Conjecture Fieldbook maintainers

Formal expression

det⁡JF∈k×⟹?F−1∈k[x1,x2]2\det JF\in k^\times\stackrel{?}{\Longrightarrow}F^{-1}\in k[x_1,x_2]^2

Why it matters

The two-variable case is a sharply scoped question about when local polynomial invertibility forces global polynomial invertibility.

Context and known approaches

The accepted source snapshot explicitly keeps the two-dimensional case open and separates it from a solved-since-1995 row about higher dimensions.

Formal boundary

If F:k²→k² is polynomial over a characteristic-zero field and det JF is a non-zero constant, must F have a polynomial inverse?

Known partial results

Pending review. No structured partial-result note has been added to this dossier.

Source trail

  1. Wikipedia — Statement, reductions, and references.
  2. Wikipedia open-problems list — Composite index by mathematical field.
  3. Wikipedia list subsection — Placement in the accepted revision-pinned living-list snapshot.