Search finding: the -2 constant-Jacobian non-injective map is Vitushkin's classic 'fake counterexample' to the Jacobian Conjecture

SEARCH_FINDING.md

The "-2 Jacobian" non-injective map — what a search turns up

Search query: Jacobian conjecture fake counterexample constant Jacobian -2 non-injective

The hit: Vitushkin's classic example

A map with Jacobian determinant exactly −2 that is not injective is a well-known "trick" example, usually attributed to Vitushkin:

F(x, y) = ( x²y⁶ + 2xy² ,  xy³ + 1/y )
det J(F) = -2   (constant)
F(-3, -1) = F(1, 1) = (3, 2)   ⇒ not injective

The signature is identical to the map in this repo: constant Jacobian −2, several distinct points colliding. That is not a coincidence — this whole family is built around the −2 gadget.

Why it does NOT disprove the Jacobian Conjecture

Vitushkin's F is a rational map, not a polynomial one: the 1/y term means it is not defined on all of ℝ²/ℂ² (blows up on y = 0). Keller's Jacobian Conjecture is a statement about polynomial maps defined on all of ℂⁿ. A map that is undefined somewhere, or only rational, is outside the hypothesis, so it is not a counterexample. This is exactly why it circulates as a "fake counterexample."

Status of the real problem

Keller's Jacobian Conjecture (polynomial F: ℂⁿ → ℂⁿ, det J(F) ∈ ℂ* ⇒ F is a polynomial automorphism) is open — no accepted counterexample exists. The only proven "non-injective, nonvanishing Jacobian" polynomial result is Pinchuk's 1994 map, which attacks the real Jacobian conjecture (nowhere-zero, non-constant Jacobian; degrees 10 and 35) — a different statement.

Sources

• Vitushkin example, discussion of constant-Jacobian non-injective maps of ℂ²: https://arxiv.org/pdf/math/0408048 ("Two remarks on non-zero constant Jacobian polynomial map of C²")
• Pinchuk's genuine real-JC counterexample: https://arxiv.org/pdf/1202.2949
• Survey / intro: https://math.uchicago.edu/~may/REU2018/REUPapers/Garland.pdf

Note on the map in this repo

The map here F: ℂ³ → ℂ³ is polynomial (no denominators), and CAS checks confirm det J(F) ≡ -2 and three genuine distinct preimages of (-1/4, 0, 0). Those computations are correct. But "polynomial + constant Jacobian + non-injective over all of ℂ³" would be a genuine refutation of an 80+ year open problem, so it must be treated as an extraordinary claim requiring peer review, not a settled result. If it survives that, it is a publishable theorem, not a gist.