On disproving and resurrecting the Jacobian conjecture (1939)
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Ruud H.G. van Tol
Jul 20, 2026, 7:12:32 AMJul 20
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https://officechai.com/ai/an-anthropic-researcher-says-fable-just-helped-him-disprove-the-85-year-old-jacobian-conjecture/
See also A021009.
"Tsuchimoto in 2005, and independently Belov-Kanel and Kontsevich in
2007, showed that the Dixmier conjecture is stably equivalent to the
Jacobian conjecture."
--- Ruud
Geoffrey Caveney
Jul 20, 2026, 12:17:25 PMJul 20
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Very interesting! The name of the mathematician who found and posted the result is Levent Alpoge. It is striking that the counterexample polynomial and points have relatively small and simple coefficients and coordinates:
"((1+xy)^3 z + y^2 (1+xy) (4+3xy), y + 3 x (1+xy)^2 z + 3 x y^2 (4+3xy), 2 x - 3 x^2 y - x^3 z): \C^3\to \C^3, has jacobian determinant -2, and sends (0, 0, -1/4), (1, -3/2, 13/2), and (-1, 3/2, 13/2) to (-1/4, 0, 0)" [quoting Alpoge's tweet]
Or as mathematician Jared Duker Lichtman described it in his own tweet commenting on the discovery:
"F(x,y,z) = ((1+xy)^3 z + y^2 (1+xy) (4+3xy), y + 3 x (1+xy)^2 z + 3 x y^2 (4+3xy), 2 x - 3 x^2 y - x^3 z)
has det(J_F) = -2 non-zero. However F is not invertible, since F sends three different pts (0, 0, -1/4), (1, -3/2, 13/2), and (-1, 3/2, 13/2) to the same image (-1/4, 0, 0)."
It is remarkable that all of the coefficients in the counterexample polynomial are <= 4 and all of the coordinates of the counterexample points are <= 13/2.
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brad klee
Jul 20, 2026, 12:29:17 PMJul 20
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The story seems to be another drop in the bucket dreams dashed
by LLM's being good at combinatorial searching. Is that right?
All I could see was a tweet, not much explanation what the prompt
was like or how the shiny counterexample was found.
Discoveries like this are helpful to young researchers avoiding
dead ends, but also I don't think it means the conjectural story
is over.
What is so special about this new example that it wasn't that
easy to find? Can we refine the conjecture on a subset? Or
is this enough to throw away the conjecture?
This cryptic statement also probably has some manifold geometry
behind it somewhere. I wonder what that looks like.
> it is remarkable that all of the coefficients in the
counterexample
> polynomial are <= 4 and all of the coordinates of the
counterexample
> points are <= 13/2.
Yes, agree. This should actually give us a complexity bound on the
space of similar polynomial structures.
Do you have an estimate if this was coded into a search space, what
the size of the search space would be?
If not I can probably get it, but I've just discovered Hadrien Brochet's
dissertation and I'm on some other exciting calculus right now.
Stay tuned...
--Brad
Christian Sievers
Jul 20, 2026, 9:26:12 PMJul 20
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Claude has been asked how one could come up with this counterexample:
(second half).
I found this in a "toot" from John Carlos Baez here:
- but right now mathstodon's web interface doesn't seem to work...
All the best
Christian
brad klee
Jul 20, 2026, 11:44:14 PMJul 20
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Another Claude's word salad... For stuff like this, I also try asking to remove
jargon and explain in lay person's terms. The symmetry condition is insightful,
and maybe the historical info is also right. I had to filter a lot to get that.
Claude gave me some wacko nonsense when I asked about search bounds, and
to be more fair, my best guess with pencil and paper probably wouldn't be much
better. Maybe the tweet author will enlighten us later? Anyone else?
If all we get is a certificate, the personal stories angle is easier to follow up on.
What I read about Yitang Zhang did not sound good for Purdue in the late '80s,
but they must have had at least one or two good people around.
Here's also a more recent article interviewing Yitang Zhang who recently
moved back to China :
It's an interesting read, but perhaps not the most candid. Is the answer
about music evasive regarding his interest in poetry?
Dr. Yitang Zhang also pledges to help his students with job searches.
That's a very noble and unique stance. Most of what I ever heard was:
"We can help you get an education, but we can't help you get a job".
All the best,
--Brad
Fred Lunnon
Jul 21, 2026, 8:03:15 PMJul 21
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Terry Tao has just put a neat summary of this matter on his "What's new" WordPress site. WFL
_
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Ruud H.G. van Tol
Jul 22, 2026, 7:19:21 AMJul 22
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On 2026-07-22 02:02, Fred Lunnon wrote:
> Terry Tao has just put a neat summary of this matter on his "What's
> new" WordPress site. WFL
-- Ruud
Ruud H.G. van Tol
Jul 22, 2026, 7:32:47 AMJul 22
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On 2026-07-22 02:02, Fred Lunnon wrote:
> Terry Tao has just put a neat summary of this matter on his "What's
> new" WordPress site. WFL
https://terrytao.wordpress.com/2026/07/21/a-digestion-of-the-jacobian-conjecture-counterexample/
> new" WordPress site. WFL
which mentions:
https://link.springer.com/article/10.1007/BF02674884 (A. G. Vitushkin, 1999)
https://sbseminar.wordpress.com/2009/05/27/how-not-to-prove-the-jacobian-conjecture/
https://sbseminar.wordpress.com/2026/07/20/the-new-counterexample-to-the-jacobian-conjecture/
-- Ruud
White Board
Jul 22, 2026, 8:25:43 AMJul 22
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The paper cited in A021009 says that both conjectures are equivalent and how to construct a counter-example of Dixmier conjecture from a counter-example of the Jacobian conjecture. Also the comments found here indicate that "Alpöge’s counterexample also refutes the Dixmier conjecture for A_n, n>=3.
--Rad
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sven-h...@gmx.de
Jul 26, 2026, 7:04:58 AM (10 days ago) Jul 26
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Hello,
there is another interesting link to the topic, a youtube video, unfortunately in German, where someone explains the counter example in a very easy way, so that everybody can understand (in German). Perhaps one can understand what he is telling just from visible formula content he is showing - without understanding the language.
https://www.youtube.com/watch?v=GcpZHXVEaQE
He just explains that the determinant of the polynom rR3->R3 is constant -2 and there are 3 input points that have the same point as result.
So you cannot calculate backwards from the result at this point.
Just found this video much easier to understand than Terence Tao post. The conjecture was in a list of 20 most difficult problems, which includes Riemanns conjecture as first entry. (List at min 4:25). And there were some more conjectures depending on Jacobis conjecture.
At min 11:15 he mentions a Russian paper from 1999 with Jakobi conjecture in the title. There was a function R2->R2 as example with constant determinant -2. The function was not a polynom, it had a term 1/y at one place, but it looks a bit similar to the current example (he says). Might be there was some construction in 3 dimensions to replace that term 1/y somehow. But even to explain that would be above my skills. Unfortunately the AI does not tell its way of construction the counter example.
Sven
-----Ursprüngliche Nachricht-----
Von: 'Ruud H.G. van Tol' via SeqFan <seq...@googlegroups.com>
Gesendet: Mittwoch, 22. Juli 2026 13:33
An: seq...@googlegroups.com
Betreff: Re: [SeqFan] On disproving and resurrecting the Jacobian conjecture (1939)
there is another interesting link to the topic, a youtube video, unfortunately in German, where someone explains the counter example in a very easy way, so that everybody can understand (in German). Perhaps one can understand what he is telling just from visible formula content he is showing - without understanding the language.
https://www.youtube.com/watch?v=GcpZHXVEaQE
He just explains that the determinant of the polynom rR3->R3 is constant -2 and there are 3 input points that have the same point as result.
So you cannot calculate backwards from the result at this point.
Just found this video much easier to understand than Terence Tao post. The conjecture was in a list of 20 most difficult problems, which includes Riemanns conjecture as first entry. (List at min 4:25). And there were some more conjectures depending on Jacobis conjecture.
At min 11:15 he mentions a Russian paper from 1999 with Jakobi conjecture in the title. There was a function R2->R2 as example with constant determinant -2. The function was not a polynom, it had a term 1/y at one place, but it looks a bit similar to the current example (he says). Might be there was some construction in 3 dimensions to replace that term 1/y somehow. But even to explain that would be above my skills. Unfortunately the AI does not tell its way of construction the counter example.
Sven
-----Ursprüngliche Nachricht-----
Von: 'Ruud H.G. van Tol' via SeqFan <seq...@googlegroups.com>
Gesendet: Mittwoch, 22. Juli 2026 13:33
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Betreff: Re: [SeqFan] On disproving and resurrecting the Jacobian conjecture (1939)
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brad klee
Jul 26, 2026, 8:39:58 AM (10 days ago) Jul 26
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Second: A blog post with a lot of jargon and no pictures doesn't do
much to help understand a geometry problem.
I'd really like to see a circle through three points taken through
the map into three lobes, and maybe that could be the starting
place for a few arclength calculations?
In all of this talk, talk, talk I still haven't seen that, but
admittedly all I did was scan through thumbnails looking for
a graph picture.
I asked Harm.On.ica to try and do it but because Harm.On.ica
doesn't have eyes, the result turned out not to have a good
view angle.
Steve is working on some vis. tools and his prototypes look pretty
good so maybe he will enlighten us someday soon.
What Harm.On.ica S-O-L, yes, sir! 5.6 TM Open AI can do:
Read the youtube transcript (using whisper.cpp? Or some trade
secret speech-to-text from google?) and condense talking points.
I've included an auto-gen summary from translation below.
Anyways, this is a lot of hullaballoo about one tweet, where
the author has positioned himself like a zen master I guess.
I'll look forward to the paper.
All the best,
--Brad
Main talking points:
1. Levent Alpoge reportedly posted an explicit polynomial map that appears
to disprove the general Jacobian Conjecture. Its Jacobian determinant
is the constant -2, but several distinct points are mapped to the same
point, so the map is not invertible.
2. The Jacobian Conjecture says that a polynomial map from n-dimensional
space to itself with a nonzero constant Jacobian determinant should
have a polynomial inverse.
3. The conjecture was formulated by Ott-Heinrich Keller in 1939. It became
a famous unsolved problem and attracted many attempted proofs, several
of which were later found to contain errors.
4. The claimed counterexample is important because it can be checked
directly. A computer algebra system can verify that the Jacobian
determinant is -2 and that the listed points have the same image.
5. Artificial intelligence reportedly played a role in finding the
example. Alpoge used Anthropic's Claude, but the video stresses that
Alpoge is a highly trained mathematician capable of asking the right
questions and verifying the result.
6. The rapid announcement caused a debate about verification and
publication. Some people argued that the example could be accepted
immediately because anyone could check it, while others wanted a
formal publication or established secondary source.
7. The construction may be related to an obscure 1999 Russian paper
containing a similar two-variable map. That earlier example was not
polynomial because it involved division. Introducing a third variable
may remove the division and produce the polynomial counterexample.
8. This raises questions about AI attribution. It is unclear whether the
AI independently reconstructed the idea or drew on obscure material in
its training data without being able to identify or cite the source.
9. The example works in dimension 3 and can be extended to every higher
dimension by adding unused variables. The one-dimensional case is
elementary and true.
10. The two-dimensional, or planar, Jacobian Conjecture remains open. The
video presents this as the main surviving and possibly most
interesting version of the problem.
11. The overall video combines a mathematical explanation, a history of
the conjecture, an account of the claimed counterexample, and a
discussion of how AI-assisted mathematical discoveries should be
verified and credited.
much to help understand a geometry problem.
I'd really like to see a circle through three points taken through
the map into three lobes, and maybe that could be the starting
place for a few arclength calculations?
In all of this talk, talk, talk I still haven't seen that, but
admittedly all I did was scan through thumbnails looking for
a graph picture.
I asked Harm.On.ica to try and do it but because Harm.On.ica
doesn't have eyes, the result turned out not to have a good
view angle.
Steve is working on some vis. tools and his prototypes look pretty
good so maybe he will enlighten us someday soon.
What Harm.On.ica S-O-L, yes, sir! 5.6 TM Open AI can do:
Read the youtube transcript (using whisper.cpp? Or some trade
secret speech-to-text from google?) and condense talking points.
I've included an auto-gen summary from translation below.
Anyways, this is a lot of hullaballoo about one tweet, where
the author has positioned himself like a zen master I guess.
I'll look forward to the paper.
All the best,
--Brad
Main talking points:
1. Levent Alpoge reportedly posted an explicit polynomial map that appears
to disprove the general Jacobian Conjecture. Its Jacobian determinant
is the constant -2, but several distinct points are mapped to the same
point, so the map is not invertible.
2. The Jacobian Conjecture says that a polynomial map from n-dimensional
space to itself with a nonzero constant Jacobian determinant should
have a polynomial inverse.
3. The conjecture was formulated by Ott-Heinrich Keller in 1939. It became
a famous unsolved problem and attracted many attempted proofs, several
of which were later found to contain errors.
4. The claimed counterexample is important because it can be checked
directly. A computer algebra system can verify that the Jacobian
determinant is -2 and that the listed points have the same image.
5. Artificial intelligence reportedly played a role in finding the
example. Alpoge used Anthropic's Claude, but the video stresses that
Alpoge is a highly trained mathematician capable of asking the right
questions and verifying the result.
6. The rapid announcement caused a debate about verification and
publication. Some people argued that the example could be accepted
immediately because anyone could check it, while others wanted a
formal publication or established secondary source.
7. The construction may be related to an obscure 1999 Russian paper
containing a similar two-variable map. That earlier example was not
polynomial because it involved division. Introducing a third variable
may remove the division and produce the polynomial counterexample.
8. This raises questions about AI attribution. It is unclear whether the
AI independently reconstructed the idea or drew on obscure material in
its training data without being able to identify or cite the source.
9. The example works in dimension 3 and can be extended to every higher
dimension by adding unused variables. The one-dimensional case is
elementary and true.
10. The two-dimensional, or planar, Jacobian Conjecture remains open. The
video presents this as the main surviving and possibly most
interesting version of the problem.
11. The overall video combines a mathematical explanation, a history of
the conjecture, an account of the claimed counterexample, and a
discussion of how AI-assisted mathematical discoveries should be
verified and credited.
jpallouche.math
Jul 26, 2026, 9:50:27 AM (10 days ago) Jul 26
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Thanks!
Note that a (possibly too simplified?) youtube link in English is
available at https://www.youtube.com/watch?v=8OBxOuyDBU8
best
jean-paul
brad klee
Jul 26, 2026, 11:05:11 AM (10 days ago) Jul 26
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> Note that a (possibly too simplified?) youtube link in English is
Or maybe too cool-ified?
Although this video draws my attention, I'm not sure that the tube
geometry at "&t=140s" (or 2:20, of course) is a literal graph of
these functions, which have caused such a stir.
Getting back to my circle idea, it would be easy to get the circular
arclength in the domain using arctan, right?
That sounds plebeian, but has anyone done that?
Oh, I know! I can ask Harm.On.ica!
<<
Use these directly in OEIS search:
R:
3.615740740740740740740740740741
alpha:
0.521972516127961676191872944161
short arc s23:
3.774634584221648788017155272125
long arc s12 = s31:
9.471867256285603935803109782184
circumference:
22.718369096792856659623374836493
Digits after the decimal point only:
615740740740740740740740740741
521972516127961676191872944161
774634584221648788017155272125
471867256285603935803109782184
718369096792856659623374836493
>>
This is also a type of thing that Lean should be able to
verify very easily, right?
Thanks for your time,
--Brad
Victor Miller
Jul 26, 2026, 5:54:33 PM (10 days ago) Jul 26
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The Jacobian conjecture was essentially a local to global principle
similar to the Hasse criterion for quadratic forms. But the experience
in Number Theory shows that it isn't always true, so I'm not surprised
that it's false.
similar to the Hasse criterion for quadratic forms. But the experience
in Number Theory shows that it isn't always true, so I'm not surprised
that it's false.
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brad klee
Jul 27, 2026, 12:34:42 PM (9 days ago) Jul 27
to SeqFan
You can take a flat isosceles triangle, re-scale it into equilateral, cut it
out of the plane, and wrap it around so that the three corners touch.
out of the plane, and wrap it around so that the three corners touch.
That's kind of what I've done in the image below. The surprising part
is how simple the algebraic realization is.
is how simple the algebraic realization is.
That's kind of what the Hodge conjecture is about, isn't it? There
are a lot of algorithms in algebraic geometry that we know will do
something, but closure and halting might not be guaranteed.
are a lot of algorithms in algebraic geometry that we know will do
something, but closure and halting might not be guaranteed.
The pre-image geometry is one of the Ramanujan elliptic curves,
so I've asked Harm.On.ica S-O-L to try and pull back the area
integrals and see if there's anything interesting.
so I've asked Harm.On.ica S-O-L to try and pull back the area
integrals and see if there's anything interesting.
We're more like to get just a mess, but iirc, if the pullback is nice
enough we just have a linear combinations problem. I'm not the
best theorist, so don't quote me on that!
enough we just have a linear combinations problem. I'm not the
best theorist, so don't quote me on that!
The picture above looks pretty cool though, something else we
need to check soon.
need to check soon.
All the best,
sven-h...@gmx.de
Jul 28, 2026, 10:16:43 AM (8 days ago) Jul 28
to seq...@googlegroups.com
Hello Brad,
interesting image, is the point in the middle the one where different points are mapped to one resulting point ? And what software did you use for the graphic?
Thanks
Sven
Von: 'brad klee' via SeqFan <seq...@googlegroups.com>
Gesendet: Montag, 27. Juli 2026 18:35
An: SeqFan <seq...@googlegroups.com>
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Marc LeBrun
Jul 28, 2026, 12:20:38 PM (8 days ago) Jul 28
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Heh, for those who prefer non-AI methods, there's this:
 | |
brad klee
Jul 28, 2026, 1:08:28 PM (8 days ago) Jul 28
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> interesting image, is the point in the middle the one where
> different points are mapped to one resulting point ?
> And what software did you use for the graphic?
Response from Harm.On.ica S-O-L 5.6:
> I’ll take that one: yes, the middle point is the common image
> of several distinct source points, and the graphic was generated
> in Python/JupyterLab with NumPy and Plotly—you can view the
> code on GitHub at
? and launch the live version on Binder at
> No capitalists were harmed in the production of this document.
It looks like On.ica has been studying Hamilton-Abel theory, and
has now learned to use Abel-Wick rotations. It's easier to see that
blue + yellow passes through the green center. I can't really tell
about the other two centers, the yellow and blue curves are folded
over pretty bad. (image inline below)
All the bes,t
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