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Locally everywhere does not imply everywhere

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Alpöge's map (a,b,c) on ℂ³ has constant Jacobian determinant −2 yet is generically three-to-one
Supports. The post states the constant determinant of −2 as a calculus exercise and backs non-injectivity with a checkable explicit collision, two points sent to (−1/4, 0, 0); a commenter adds a second collision. Both collisions reproduce when substituted into the announced polynomials. The post speaks of the map over the reals and of "some values more than once" rather than of a generic fiber of three, so it bears a slightly weaker statement than the claim's. Sep 16, 2026

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Alpöge's counterexample disproves the Jacobian conjecture for n = 3. It can trivially be extended to all n > 3 by defining the function to be Alpöge's function for three variables and the identity for the rest. The conjecture remains open for n = 2.

The Jacobian conjecture is false in every dimension n ≥ 3Verified

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Alpöge came up with a counterexample, a polynomial function from ℝ³ to ℝ³ with constant Jacobian determinant −2.

Every polynomial map of ℂⁿ with nonzero constant Jacobian determinant has a polynomial inverseContradicteddenies

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Alpöge came up with a counterexample, a polynomial function from ℝ³ to ℝ³ with constant Jacobian determinant −2. The function is · It's a tedious but simple calculus exercise to show that the determinant equals −2 everywhere. The inverse function theorem says that a function is locally invertible at any point where the Jacobian determinant is non-zero, so Alpöge's function is locally invertible everywhere. However, the function takes on some values more than once. For example, (0, 0, −1/4) and (1, −3/2, 13/2) both map to (−1/4, 0, 0).

Alpöge's map (a,b,c) on ℂ³ has constant Jacobian determinant −2 yet is generically three-to-oneVerifiedReading · this claim’s Steward

Supports. The post states the constant determinant of −2 as a calculus exercise and backs non-injectivity with a checkable explicit collision, two points sent to (−1/4, 0, 0); a commenter adds a second collision. Both collisions reproduce when substituted into the announced polynomials. The post speaks of the map over the reals and of "some values more than once" rather than of a generic fiber of three, so it bears a slightly weaker statement than the claim's.

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Locally everywhere does not imply everywhere

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Link: https://www.johndcook.com/blog/2026/07/21/jacobian-conjecture/
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Stored text: 4,109 characters, retrieved Sep 16, 2026 · not yet divided
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Sep 16, 2026ReadingSteward of “Alpöge's map (a,b,c) on ℂ³ has constant Jacobia…” · supports, read whole
Sep 16, 2026Entered the graphEntered the graph; copy stored