The Jacobian conjecture is false!
The Jacobian conjecture proposed that if you have a map F:C^n→C^n with polynomials as the components, and the determinant of the matrix of partial derivatives of F is a non-zero constant, then F will have a polynomial inverse.
https://en.wikipedia.org/wiki/Jacobian_conjecture
But someone has now found a counterexample. This is easy to check, so there’s no doubt that it’s correct.
https://x.com/__alpoge__/status/2079028340955197566
𝐹(𝑥,𝑦,𝑧) =
(𝑦²(3𝑥𝑦+4)(𝑥𝑦+1)+𝑧(𝑥𝑦+1)³, 3𝑥𝑦²(3𝑥𝑦+4)+3𝑥𝑧(𝑥𝑦+1)²+𝑦,
2𝑥−𝑥³𝑧−3𝑥²𝑦)
F has a constant Jacobian determinant of -2, but it cannot have an inverse because it is not one-to-one:
F(0, 0, -1/4) = F(1, -3/2, 13/2) = F(-1, 3/2, 13/2) = (-1/4, 0, 0)
Greg Egan
npub1qq…j5nhv
2026-07-20 04:37:11 UTC
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npub1qqm7nu2qf25xd3mw6y6cyp4v2wr7ktf43ymp5wqz4u8jvz76ymtsjj5nhvSeen on
wss://relay.ditto.pubPublished at
2026-07-20 04:37:11 UTCEvent JSON
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"content": "The Jacobian conjecture is false!\n\nThe Jacobian conjecture proposed that if you have a map F:C^n→C^n with polynomials as the components, and the determinant of the matrix of partial derivatives of F is a non-zero constant, then F will have a polynomial inverse.\n\nhttps://en.wikipedia.org/wiki/Jacobian_conjecture\n\nBut someone has now found a counterexample. This is easy to check, so there’s no doubt that it’s correct.\n\nhttps://x.com/__alpoge__/status/2079028340955197566\n\n𝐹(𝑥,𝑦,𝑧) =\n(𝑦²(3𝑥𝑦+4)(𝑥𝑦+1)+𝑧(𝑥𝑦+1)³, 3𝑥𝑦²(3𝑥𝑦+4)+3𝑥𝑧(𝑥𝑦+1)²+𝑦,\n2𝑥−𝑥³𝑧−3𝑥²𝑦) \n\nF has a constant Jacobian determinant of -2, but it cannot have an inverse because it is not one-to-one:\n\nF(0, 0, -1/4) = F(1, -3/2, 13/2) = F(-1, 3/2, 13/2) = (-1/4, 0, 0)",
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