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Conjecture

Jacobian Conjecture

Algebraic Geometry

The Jacobian conjecture is a conjecture concerning polynomials in several variables that states that if a polynomial function from an n-dimensional space to itself has a Jacobian determinant that is a non-zero constant, then the function has a polynomial inverse. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/

Facts
Statement
If a polynomial function from an n dimensional space to itself has a Jacobian determinant that is a non zero constant, then the function has a polynomial inverse. 2
Proposed Year
1939 3
Progress Toward Resolution
Disproved for three or more variables: on 2026-07-19 Levent Alpoge presented an explicit counterexample in three variables, credited to an AI model, and a counterexample in three variables gives counterexamples in every higher dimension. The case of two variables (the planar Jacobian conjecture) remains open, and the case of one variable is trivially true. 3
Classification
Resolution Status
Disproven 1
Chronology
Resolved Year
2026 1
Prize Status
Prize Status (category)
No Prize Offered 2
Connections

In Branch

Field Theory, Branches of Mathematics
Source Wolfram MathWorld: Jacobian Conjecture

Posed By

Ott-Heinrich Keller, Mathematicians
Source Jacobian Conjecture (Wikipedia)
Sources
1. A digestion of the Jacobian conjecture counterexample (Terence Tao, What's new, 21 July 2026)
  • Opening paragraph, first sentence
    It was recently shown (using the Fable AI) that the conjecture is false in three dimensions (and thus in higher dimensions as well)
  • Opening paragraph, closing sentence
    The conjecture remains open in two dimensions, and is easy to establish in one dimension.
View the Source
2. Jacobian Conjecture (Wikipedia)
  • Lead section, formal statement sentence
    In mathematics, the Jacobian conjecture is a conjecture concerning polynomials in several variables that states that if a polynomial function from an n-dimensional space to itself has a Jacobian determinant that is a non-zero constant, then the function has a polynomial inverse.
  • Lead section
View the Source
3. Jacobian Conjecture (Wikipedia)
Wikimedia Foundation
  • Lead section
    if a polynomial function from an n-dimensional space to itself has a Jacobian determinant that is a non-zero constant, then the function has a polynomial inverse
  • History section, Keller sentence
    the modern version of the Jacobian conjecture in n dimensions was formulated by Ott-Heinrich Keller in 1939 for the case of polynomials with integer coefficients.
  • Results section, counterexample sentence
    presented an explicit counterexample to the conjecture in three-dimensional space which he credited to Claude Fable 5 AI model.
  • Results section, planar case sentence
    The case n = 2 (two variables), also called the plane Jacobian conjecture or planar Jacobian conjecture, is the only case that remains unresolved as of 2026.
  • Posed By: Ott-Heinrich Keller, History
    was formulated by Ott-Heinrich Keller in 1939 for the case of polynomials with integer coefficients
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Wolfram MathWorld: Jacobian Conjecture
Wolfram MathWorldIn Branch: Field Theory, Subject classifications
Quote, In Branch: Field Theory, Subject classifications
Algebra Field Theory
View the Source
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