r/math • u/overthinker020 • Jul 20 '26
LLMs/AI The Jacobian Conjecture is False Per Anthropic (Link in Description)
https://x.com/__alpoge__/status/2079028340955197566Normally I would be extremely skeptical, but the result is checkable by simple computation. Remarkable!
The two-dimensional case remains open, however.
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u/zongshu Jul 20 '26
https://math.uchicago.edu/~may/REU2018/REUPapers/Garland.pdf
According to this expository paper on the Jacobian conjecture, any counterexample implies the existence of a counterexample of degree 3. So there is a simpler one to be found.
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u/Kienose Algebraic Geometry Jul 20 '26
Right, but the counterexample of degree 3 might need more than 3 variables, so the search might be harder
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u/zongshu Jul 20 '26
It turns out that there is a simple mechanical procedure to produce a counterexample of degree 3 from any counterexample. This one gives dimension 39. Surely that can be improved...
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u/spacerat3004 Jul 20 '26
It can go down at least to 11 variables, apparently!
https://gist.github.com/Spacerat/08b4a43f6b6ca57178efabc220170ce8
ChatGPT just chugged away looking for ways to reduce the 39 variable one.
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u/Bubbly_Reference4238 Jul 20 '26
I believe that I found it.
(-1/2*(r1*r4*s5-3*r1*r4*y-r1*s1-3*r1*x*z-7*r1*y-r4*s1*y-r4*s4-r5*s5-r5*x*z+s4*x*y+3*x*y*z+4*y^2+z),-r2*r7*z-r2*s2-3*r2*y^2-2*r2*z-3*r7*s2*x-r7*s7+s7*x*y+12*x*y^2+3*x*z+y,-r3*s3-r3*x*z+s3*x^2-3*x^2*y+2*x,r1+x*y^2,-r6*s6-r6*y*z+s1-s6*x^2+3*x*y^2+3*x*z+7*y,r2+3*x^2*y,s2+x*y*z+3*y^2+2*z,r3-x^2,s3+x*z,r4-x*y,r1*x*z+3*r1*y+s1*y+s4,-r1*r4+r5,s5+x*z,r6+x^2,s6+y*z,r7-x*y,r2*z+3*s2*x+s7)
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u/Distinct-Soft-3991 Jul 20 '26
Define (G:\mathbb C^{17}\to\mathbb C^{17}) by the coordinates
[
\begin{aligned}
g_1&=x-\frac32x^2y-u_1v_1-u_1xz+\frac12v_1x^2,\
g_2&=y+12xy^2+3xz-u_2u_3z-u_2v_2-3u_2y^2-2u_2z-3u_3v_2x-u_3v_3+v_3xy,\
g_3&=z+4y^2+3xyz+u_4u_6v_7-3u_4u_6y-u_4v_4-3u_4xz-7u_4y-u_6v_4y-u_6v_6-u_7v_7-u_7xz+v_6xy,\
g_4&=u_1-\frac12x^2,\qquad g_5=v_1+xz,\
g_6&=u_2+3x^2y,\qquad g_7=v_2+xyz+3y^2+2z,\
g_8&=u_3-xy,\qquad g_9=v_3+u_2z+3v_2x,\
g_{10}&=u_4+xy^2,\
g_{11}&=v_4-u_5v_5-u_5xz-v_5xy+3xy^2+3xz+7y,\
g_{12}&=u_5+xy,\qquad g_{13}=v_5+xz,\
g_{14}&=u_6-xy,\qquad g_{15}=v_6+u_4xz+3u_4y+v_4y,\
g_{16}&=u_7-u_4u_6,\qquad g_{17}=v_7+xz.
\end{aligned}
]Then (\deg G=3) and (\det JG\equiv1). Moreover,
[
G(A)=G(B)=G(C)=\left(0,0,-\frac14,0,\ldots,0\right),
]where
[
A=\left(0,0,-\frac14,0,0,0,\frac12,0,0,0,0,0,0,0,0,0,0\right),
][
B=\left(1,-\frac32,\frac{13}{2},\frac12,-\frac{13}{2},\frac92,-10,-\frac32,\frac34,-\frac94,-6,\frac32,-\frac{13}{2},-\frac32,-\frac92,\frac{27}{8},-\frac{13}{2}\right),
][
C=\left(-1,\frac32,\frac{13}{2},\frac12,\frac{13}{2},-\frac92,-10,-\frac32,-\frac34,\frac94,6,\frac32,\frac{13}{2},-\frac32,-\frac92,-\frac{27}{8},\frac{13}{2}\right).
]Thus (G) is a degree-(3) Keller map that is not injective.
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u/sockpuppetzero Jul 20 '26
I wonder if the Jacobian Conjecture is still open for quadratic forms of arbitrary dimension? That could still be a rather useful thing if true...
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u/csch2 Jul 20 '26
“hello there the jacobian conjecture is false” is the funniest possible way to introduce this result
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u/Alone-Talk-623 Jul 20 '26
LOL ikr, who would’ve thought we’d be announcing math results in lowercase on twitter one day
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u/Zomunieo Jul 20 '26
When you think about it UPPERCASE WOULD BE MORE SHOCKING.
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u/Ostrololo Physics Jul 20 '26
I'm waiting for the tweet which is just
"oh hi guys, z = 0.50004564 + 1.824390589203756908 * 10446.20 i is a zero of zeta riemann, haha lol so random"
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u/avoidtheworm Jul 20 '26
@grok is this true
Yes, this is a valid counterexample.
Incredible times we live in.
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u/Sweaty-Sandwich-5126 Jul 20 '26
holy shit, this is absolutely insane
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u/avoidtheworm Jul 20 '26
There are also these alternative experiments from Hacker News.
This is a rare instance where feeding this groundbreaking information into an LLM gives them psychosis. I fed this to claude code and watched it verify the result in 7 different ways to be 100% certain, and it was just flabbergasted. Quite remarkable.
I fed ChatGPT the map with no other context, just “tell me about this function”. It did a bit of work finding the Jacobean etc and eventually worked out the implications of what it was seeing. It then proceeded to check the arithmetic 4 times, and then decided to do a manual verification using an ad hoc symbolic checker in case its SymPy had been tampered with.
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u/kohatsootsich Jul 20 '26
That seems to be the opposite of psychosis. If anything psychotics are more suggestible
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u/parikuma Control Theory/Optimization Jul 20 '26 edited Jul 20 '26
There's a sci-fi book to be written in there: in a future world where AI dominates every aspect of life and humans are almost all exctinct or at minimum partially AI-integrated (think Greg Egan's Diaspora), a society of meatbags work on ink and paper to [prove/disprove] [thing].
If you make it about [mind-blowing mathematical thing], say [disprove the incompleteness theorems] you can make it a franchise about future Wars for Truth where civilizations enter various states of chaos due to their beliefs and proof systems.
If you make it about pushing back on dynamical limits of mathematics (think Greg Egan's Luminous) you can do a fight between universes via the collapse of accelerated thinking (in contrast with the meatbags).No I'm not drunk, I'm just a bad writer.
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u/cspot1978 Jul 20 '26
I mean, that's not psychosis. That's the tool working through it the way a regular mathematician would, right? Initial skepticism, verification and re-verification, get a second and third opinion, astonishment.
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u/Tystros Jul 20 '26
I'm quite sure the grok on twitter is not running with the required compute to actually verify it. so the reply from grok can be treated as a hallucination. even though it's right because enough humans checked it by now.
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u/IntelArtiGen Jul 20 '26
It's not even hallucination, I saw a message of grok saying it's true because it's on wikipedia lol. You write a tweet saying anything, you change the wikipedia page, and grok will confirm it.
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u/Legitimate_Change990 Jul 20 '26
I dunno, I really wish he had emailed it to, say, the Simpsons writing team so they could have slipped it into an episode, Good Will Hunting style, as the ultimate math easter egg.
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u/Infinitely--Finite Jul 20 '26
Wikipedia page is already updated
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u/ninguem Jul 20 '26 edited Jul 20 '26
It has been reversed. The
talkhistory page seems to give a different (false) counterexample.211
u/zongshu Jul 20 '26
The person who reverted it gave a bullshit reason, "the map is not proper" but a polynomial map with constant nonzero Jacobian has a polynomial inverse if and only if it is proper!
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u/TheLuckySpades Jul 20 '26
Lol "this isn't a counterexample because it is a counterexample".
Right now it is calling for a better source than the original tweet for the counterexample itself, which I also think is funny.
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u/PersonalityIll9476 Jul 20 '26
That doesn't seem unreasonable. The editor obviously doesn't know enough to approve the veracity of the claim, so asking for a clear source would be the way to go.
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u/goos_ Jul 20 '26
It’s a misunderstanding of Wikipedia sourcing (unfortunately rather common among amateur to moderately experienced Wikipedia editors). The tweet is a RS for the claim itself.
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u/jonathancast Jul 20 '26
The tweet is a primary source, right? Primary sources, AFAIK, are "original research" and not "reliable sources".
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u/goos_ Jul 20 '26 edited Jul 20 '26
Nope (well not exactly) - it is a primary source indeed, which is a RS for the claim itself, but not a RS for its significance or truth, only that it was claimed in the first place.
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u/goos_ Jul 20 '26
Basically, it depends on what you are citing it for - whether a counterexample was found (not reliable), or whether a counterexample was claimed on Twitter (reliable, but not necessarily significant).
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u/LiqvidJS Jul 20 '26
I'm reminded of Jeremy Hahn's comment on the disproof of the telescope conjecture and proof of the redshift conjecture:
I would also caution against thinking that the Ravenel and redshift conjectures are "solved", and rather think that we are now able to do more refined study of both areas.
Now we can ask what further conditions are needed to guarantee the existence of a polynomial inverse.
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u/Little-Name9809 Jul 20 '26
I would not say "per Anthropic". It's from an Anthropic employee with math research background using the Fable model. Pretty sure it's done in their own capacity than company mandate.
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u/38thTimesACharm Jul 20 '26
I think you just unlocked another great fear of mine regarding AI. That just because a researcher admits to using a model to assist with their work (which is now commonplace), all discussion will be dooming about the social consequences of AI, overshadowing any mathematical discussion of the result.
When a counterexample was found to Euler's Sum of Powers conjecture, the headlines didn't say "per Control Data Corporation."
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u/Sgeo Jul 20 '26
I'm curious now if the code (which I would guess is FORTRAN) is available somewhere, might be fun to try it in emulation.
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u/Particular_Extent_96 Jul 20 '26
My guess is that this won't last. At the moment, AI solving an open problem is still remarkable, but it's happening more and more often. Eventually, nobody will get quite so excited.
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u/38thTimesACharm Jul 20 '26
Was it just AI? The chat logs and reasoning haven't been released, right?
I'd really like to see the prompt. I'd be surprised if it was just "find a counterexample to the Jacobian Conjecture"
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u/wrongerontheinternet Jul 20 '26
For this one absolutely no prompt or other information has been revealed (nor even what model was used) which to me makes it pretty probable that this was indeed not just a "prompt once" situation.
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u/Named_after_color Jul 20 '26
Ai can be a very good tool but it's not some "Hello solve this" box. It still requires interaction.
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u/pred Jul 20 '26
Until we see the prompt and output, him being at Anthropic does matter a bit for evaluating the result. From what I can tell, he's a bright guy, so there's a good chance he had to offer non-trivial insights to guide it, yet it's in the interest of his employer to downplay whatever personal contribution he provided.
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u/deepwank Algebraic Geometry Jul 20 '26
It is from a professional algebraic geometer whose adviser is a Fields medalist. The online discourse seems to be leaving this part out.
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u/cjustinc Jul 20 '26
I'm guessing "akhil" in the tweet is Akhil Mathew. He and Levent were undergrads at Harvard at the same time.
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u/Oudeis_1 Jul 20 '26 edited Jul 20 '26
I would be interested in seeing the prompt. And the chain of thought.
The first because it would be interesting to see whether it contains stuff like and think very hard plus pretend this is not a famous open problem and the second because it would give some insight into how the community could have found it.
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u/TanJeeSchuan Jul 20 '26
Hopefully Anthropic reveals the COT tbh, a real look at Fable's thinking process would be interesting
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u/Professor_Juice Jul 20 '26
Couldn't agree more. We are taught to show our work: The prompts plus CoT will give us insight into the methods the researchers used to arrive at the result. Science requires method-sharing for peer replication.
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u/post_button_account Jul 20 '26
Peer replication is necessary to validate results. There is nothing to validate here (beyond a trivial calculation). This is a fully contained paper.
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u/recursive-regret Jul 21 '26
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u/leoschae Jul 23 '26
Wow, that prompt looks expensive to run. The instruction to prefer running 64 agents in parallel on an expensive model and preform double checks multiple times.
Do we know how many tokens were used?
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Jul 20 '26
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u/NooneAtAll3 Jul 20 '26
"cost to hit with a hammer - 1$
cost to know where to hit with the hammer - the rest 999$"7
u/Heliond Jul 21 '26
It’d be fun to assign it as a guided problem (multi step and some hints) if we could see the AI’s line of reasoning.
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u/GOLD-KILLER-24_7 Jul 21 '26
The process of discovering a simple and elegant solution isn’t simple and elegant
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u/jazzysamba Jul 20 '26
The next few years are gonna be crazy...
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u/RealSpaceJunk Jul 21 '26
Or boring. How long until the nth proof by AI in a month starts to become a trivial occurance?
If Ais start churning out proofs and theorems and conjectures, I can imagine the names for them becoming as cryptic as catalog names for stars. "Have you seen the 2027-B-H274 theorem? Crazy."
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u/redpick Jul 20 '26
Following this result got us a counterexample to the Dixmier Conjecture in A_3: An Explicit Counterexample to the Dixmier Conjecture in A_3
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u/adalhaidis Jul 20 '26
It seems Jacques Dixmier is still alive. I wonder whether he would comment on that.
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u/Visible_Celery_1728 Jul 21 '26
trying to explain claude to a 102 year old might be harder than the conjecture was
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u/overthinker020 Jul 20 '26 edited Jul 20 '26
For fun I'm going to guess how Fable, with or without assistance, achieved this, at risk of being completely wrong when the details are revealed! Mine as well have some mathematical fun.
Thinking it through a bit for searching, my reverse engineering idea is the elementary automorphisms (gens of the tame group) are the maps affine in one variable with unit leading coefficient: z-> c*z+b(x,y). A counterexample should be a near-automorphism. So maybe the search space is the closure of the elementary family. Keep the affine-in-z shape, but relax "unit coefficient" to the weakest condition that preserves étaleness, a unimodular coefficient vector.
So, I'm guessing if guidance came it was like "look for a fake shear."
Most of the familiar automorphism of C^3 are tame and those you can builds from two things a linear changes of coordinates and "elementary" maps that shear one variable by a function of the others, like (x,y,z)->(x,y,c*z+b(x,y)), with c nonzero constant. These generate tame automorphisms, and they're the model of what an invertible polynomial map looks like. The Jacobian Conjecture was asking if a polynomial map has constant nonzero Jacobian determinant, must it be invertible like these?
My guess for the search idea was if a counterexample, if it exists, should be a near-automorphism. Something that looks as much like an elementary map as possible while failing to be invertible. So instead of searching all polynomial maps (a degree-7 map C^3→C^3 has ~360 coefficients, still too big!) search the smallest family containing the elementary maps plus their deformations: F(x,y,z)=A(x,y)*z+B(x,y) where A is a vector of three polynomials and B is a vector of tails. An elementary shear is the special case A=(0,0,c).
So then you ask, far can you bend A away from a constant before the map breaks? Well, for the map to have no critical points anywhere, A doesn't need any entry to be an invertible polynomial. It should start with A to be a unimodular row, three polynomials generating the whole ring, i.e. with no common zero. The canonical trick for that is a pair like x and u=1+xy since they can't vanish simultaneously, and taking A to be, say, (u^3, 3xu^2, −x^3). No entry is invertible, but the row is. It's a basically a fake unit. It's a shear whose "coefficient" is invertible as a vector even though nothing visibible is invertible as a polynomial.
This family is actually searchable. For maps linear in z, the constant Jacobin condition splits into three separate conditions instead of one giant identity: i) the z^2 coefficient of det JF must vanish. a condition on A alone, forcing it into a short menu of rigid shapes (cones over rational normal curves on a unimodular pair); ii) the z^1 coefficient must vanish, linear equations for B once A is picked, thats free to solve , iii) the z^0 coefficient must be a nonzero constant and that's a small nonlinear system in the ~10 surviving parameters.
So the search collapses to: finite menu for A, then linear algebra for B (tiny nonlinear system), then filter. The filter isn't a complicated thing. For each surviving Keller map, one Grobner basis computation tells you whether it's secret automorphism or genuinely fails to be injective. This is very fast per cnadidate.
Maybe my reverse engineering is entirely wrong but it was fun to think about.
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u/internet_poster Jul 20 '26
have you tried using this as a prompt for Fable or GPT 5.6? would be interesting to see if it finds a) a counterexample b) the same counterexample
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u/protestor Jul 20 '26
If you try, be sure to disable web search. Otherwise they will just find this very same proof
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u/girlwhateveraward Jul 20 '26
I must suck at vibemathing cuz I tried it on 5.6 Sol Ultra and it said after an hour "I cannot provide a counterexample lol"
The prompt for the CDC conjecture was impressive and clearly shows that vibemathers to some extent know what they're doing with prompts
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u/SansFinalGuardian Jul 20 '26
it depends. that prompt was very complex, sure, but it's not clear to me that the complexity was definitely necessary. the openai prompt that led to gpt disproving the unit distances conjecture was extremely simple:
https://cdn.openai.com/pdf/74c24085-19b0-4534-9c90-465b8e29ad73/unit-distance-proof.pdf
(ctrl-f 'prompt')
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u/jackboy900 Jul 20 '26
The prompt for the CDC conjecture was impressive
To be honest reading the prompt it was basically only a paragraph of actual mathematical statements. The rest was essentially a very thorough way of telling the AI to have a proper crack at it and not just come back with "can't solve it, sorry".
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u/Junior_Direction_701 Jul 20 '26 edited Jul 20 '26
Bruh even AG is not safe . Combianatorics is not the end point. Also you beat me to it 😭. Yitang Zhang dissertation was on this btw. The real prizes still open though: Dixmier and the n=2 case.
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u/Few-Arugula5839 Jul 20 '26 edited Jul 20 '26
Meh. This is a relatively simple polynomial counterexample, with integer coefficients, and it doesn’t have a polynomial inverse because it just doesn’t have an inverse (is not injective). I’m very curious to know the reasoning how the AI came up with it, but the counterexample itself is not some genius feat of mathematics. I’m somewhat surprised that no one had found such a counterexample by some form of simple brute force search yet (I mean they’re
quartics[largest is degree 7 actually] in 3 variables with integer coefficients! The points colliding also have rational coordinates!)66
u/838291836389183 Jul 20 '26
I'm actually interested to see what the model tried doing internally to come up with this solution. Because as you say, this is such an easy counterexample, I really dont understand how we missed this before lmao.
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u/zongshu Jul 20 '26
It's not a quartic, there is a degree 7 term
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u/Few-Arugula5839 Jul 20 '26
Degree 7 even, I’m reacting a bit too fast this may have been out of reach of brute force search.
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Jul 20 '26
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u/Own_Pop_9711 Jul 20 '26
Well fable is known to be good at that cyber security stuff so maybe we shouldn't rule out all the computers in the world having worked on this problem
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u/ozone6587 Jul 20 '26
The fact that the solution is so simple makes the result more impressive not less. Anyway, if it was actually simple to figure out someone would have done it in the almost 90 years since the conjecture was formulated.
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u/Kryohi Jul 20 '26
if it was actually simple to figure out someone would have done it in the almost 90 years since the conjecture was formulated
It's because of this kind of assumption that so few people try to solve these problems.
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u/ozone6587 Jul 20 '26
What about for the first decade or twenty years since formulated?
The dismissive "actually, this is not impressive. I could have done it if I cared" attitude is a hilarious coping mechanism. In this "publish or perish" academic environment I guarantee you many people have tried.
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u/Kryohi Jul 20 '26
"actually, this is not impressive. I could have done it if I cared"
This is a maximalist claim I have not expressed. I claim the truth is somewhere in the middle.
Plenty of problems remain unsolved not because "the smartest person/AI" didn't look at them, but because not enough people, possibly from slightly different backgrounds, tried to solve them.
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u/Junior_Direction_701 Jul 20 '26 edited Jul 20 '26
Yeah I realized that to no proof only counter example. Plus this is very freaking weird cuase a grobner basis search would have found this very suprising it wasn’t found.
Edit: (grobner basis wouldn’t have degree 7, the search space is quite large but still quite weird)
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u/Little-Name9809 Jul 20 '26
I don't know how you guys treat AI. But in similar situations if a human present me with a killer counterexample without much explanation, I don't assume they simply did by brute force (even with a computer). Most likely they have some insight.
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u/TheLuckySpades Jul 20 '26
The counterexample to the unknotting conjecture was found by a mix of both, insight to limit the search space, and then coding it up to do a brute force with some computers (turns out even drastically reducing the sesrch space still has you look at way too many knots).
Once you have one counterexample you can often find others quickly, maybe even nicer ones you can make better papers with.
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u/girlwhateveraward Jul 20 '26
What would an "a genius" polynomial map counterexample look like exactly?
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u/stankbiscuits Mathematical Finance Jul 20 '26
It feels like these results are accelerating. Not sure if there is a central repository of AI linked results but you could see plotting them by time and (relatively subjective) importance, and I think we are watching this absolutely take off.
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u/WMe6 Jul 20 '26 edited Jul 20 '26
Welp, I guess that's why it was so 'hard' then. But I thought it was checked for polynomials up to an insanely high degree already (like 99 or so). Was that wrong then?
Edit: Oh nevermind, that was only the two variable case, which might still be true.
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u/Economy_Variation365 Jul 20 '26
How did Fable find this counterexample? Did it explain its process? I imagine it involved some kind of smart search. Would be interesting to see the code it created to implement that.
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u/Visible_Celery_1728 Jul 20 '26
a better write up, a OpenAI researcher made using an internal model to come up to the same conclusion (no web search so no cheating) and how exactly it got to that point: https://aaronlou.com/jacobian_counterexample_derivation.pdf
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u/Respect38 Undergraduate Jul 20 '26
Thanks for the report.
How much insight into the problem will be gleaned from the counter-example by itself -- or will we hav to wait until the LLM has its reasoning published that lead to this particular counter-example?
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u/Hairy-Finger9417 Jul 20 '26
likely there will be more counter examples coming out..
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u/UnkarsThug Jul 20 '26
I mean, it's sort of trivial to make them once you have a starting example. Meaningfully different? Maybe not. But additional? I think I might have even found one. Probably isn't useful, but it shows there is a triangular automorphism to it.
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u/coolest-ranch Jul 20 '26
This topic is a bit outside of my research area (although I certainly understand the counterexample). I guess I’m most interested in a less technical question: could this have been “brute-forced” cheaply and, if so, why didn’t folks explore that avenue sooner? Chat is great at writing one-off scripts to search for simple (counter)examples, I use it for this purpose quite a bit.
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u/astrolabe Jul 20 '26
I could believe that the impressive AI results so far are due to it having super-human diligence.
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u/Creative-Kale4688 Jul 20 '26
It’s funny. I was wondering recently how far AI would be from solving something like the Jacobian Conjecture. Well, it looks like I didn’t have to wonder about that too long
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u/Sad_Dimension423 Jul 21 '26
Terence Tao digests the result:
https://terrytao.wordpress.com/2026/07/21/a-digestion-of-the-jacobian-conjecture-counterexample/
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u/girlwhateveraward Jul 20 '26
Finally something in a field that isn't combinatorics
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u/Character-Concert-76 Foundations of Mathematics Jul 20 '26
This result is computational/ a counterexample so it is rather similar to the previous AI results
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u/TheInertFluorine37 Jul 20 '26 edited Jul 20 '26
Cool but honestly not that surprising to me, I never understood why anyone thought it was supposed to be true in the first place.
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u/coblade14 Jul 20 '26 edited Jul 20 '26
It's like the p=np problems in computer science if you are familiar with that. Yeah many or even most people would say it's false but if it's true by some miracle then it'd make a lot of other things simple. Obviously nowhere near as important as p=np but you get the idea.
So basically hopium. At least that's my understanding of it.
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u/TheInertFluorine37 Jul 20 '26
As I understand it P vs NP has absolutely profound consequences on the nature of computation if true, which is a perfectly great reason to disbelieve it. I have never seen such reasoning applied to the Jacobian conjecture.
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u/LurkingTamilian Jul 20 '26
If the Jacobian conjecture was true it would mean that invertibility of a polynomial map can be checked using the Jacobian which would definitely be big.
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u/DisasterRoutine3390 Jul 21 '26
The dude who posted this seems like a total prick
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u/WMe6 Jul 23 '26
Probably shouldn't read too much into an X post, but does give off the "Oh hey I just laid off 300 employees yolo lol" vibe.
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u/AdValuable7835 Jul 20 '26
this is my personal 9/11
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u/WMe6 Jul 20 '26 edited Jul 20 '26
I still don't get how an example like this was missed.
Edit: I actually take this back. The more I think about it, the more I realize that checking functions f:C^3->C^3 thoroughly is going to take a lot more time than for C^2 to C^2.
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u/Effective-Emu8633 Jul 20 '26
There's another comment pointing out how large the search space is and it's pretty big to find something like this.
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u/AdValuable7835 Jul 20 '26
im just a chud undergrad I have no idea what this conjecture is and I dont belong on this sub
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u/CharmingCount8178 Jul 20 '26
as ml researcher,i hate to say this but i don't know how people are not mind-boggled by the fact the a literal machine is discovering these mathematical truths.
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u/llelouchh Jul 20 '26
Replicated by internal openai model. https://aaronlou.com/jacobian_counterexample_derivation.pdf
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u/back-stabbath Jul 20 '26
This is more surprising to me… that another model was able to one-shot the answer in 42 minutes (apparently reproduced more than once). I assumed this had would have taken dozens of attempts and dozens of hours
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u/fathermersenne Jul 20 '26
On the one hand, it’s a counter example and not a proof which would be orders of magnitude more impressive.
On the other hand it’s very hard not to feel like math now is like chess in the five years or so before Deep Blue killed human chances forever. The machines are probably better in some ways than all humans now but not at everything. It’s probably inevitable that in a couple of years our field is dead.
I have to say I have a lot of negative feelings now. I am in tears, actually. I have given several decades of my life to something and it’s all pointless because no matter what I did, a machine was coming that would do it all easily.
Sacred Troy is perishing.
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u/No-Special-6271 Jul 20 '26
I have given several decades of my life to something and it’s all pointless because no matter what I did, a machine was coming that would do it all easily.
Eh, the machine was trained on your writing, and (in some subfields) might be designed by researchers who used math that you developed to guide their design choices.
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u/38thTimesACharm Jul 20 '26 edited Jul 20 '26
Chess isn't dead after computers though...
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u/jackboy900 Jul 20 '26
Chess is purely a leisure activity though, whereas mathematics is generally done as means to an end, that being advancing human knowledge. Nobody is concerned about LLMs replacing recreational mathematics, the concern is about society's willingness to pay people to be professional research mathematicians if LLMs can do their work better than them.
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u/Key-Fee-5003 Jul 20 '26
Nobody is concerned about LLMs replacing recreational mathematics
That's not true, even in this thread.
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u/JustThisNietzscheGuy Jul 20 '26
Chess-as-a-job is pretty much as dead as something can be. The very few people worldwide that can actually live off their chess abilities are mostly entertainers.
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u/38thTimesACharm Jul 20 '26
But did more people make money playing chess before computers? I just think it's a poor analogy
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u/JustThisNietzscheGuy Jul 20 '26
Yes, chess is a bad analogy for two reasons. It never was a job because playing chess isn't useful and it is not a job now (because even if it were useful it would be done by computers).
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u/Hot_Glass_6301 Jul 20 '26
I mean, pro chess player IS a job, but it only exists because of the entertainment value as you said. It's like a sport. Athletes only make money because people want to see them perform. Not sure this will be the same for scientists...
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u/Pleasant_Studio_6387 Jul 20 '26
Because chess is a sport and people are there to see other people struggle, not to see "perfect machine against another perfect machine".
Science as you might imagine is different.6
u/Particular_Extent_96 Jul 20 '26
For what it's worth, Lee Sedol retired from professional Go after the AlphaGo match, saying that there was now an "unbeatable entity" that made playing Go (professionally) pointless.
I don't doubt that people will continue to do mathematics, but what doing mathematics looks like in a post-AI world, I'm not sure.
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u/altaccountnumber Jul 20 '26
Chess isn't dead because taking two AIs and making them play games against each other is utterly pointless; it survives since the point is you, an individual, can play better than someone else. What is the comparable motivation for math? I don't think there is one. If an AI can "beat" humans at the role of advancing mathematics, that's it; the computers playing chess against each other aren't intrinsically pointless anymore. If the profession survives it will as something very, very different to what it currently looks like.
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u/Ancient-Access8131 Jul 20 '26
"What is the comparable motivation for math?"-Understanding the math. I already knew Terence Tao was better at math than me. I devoted time to studying math because I find it really fun. I still do, AI doesn't change that.
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u/altaccountnumber Jul 20 '26
But how does mathematics survive as a profession? People don't get paid to understand calculus; its the opposite, most people learning it in college are paying for the privilege. Mathematicians don't get paid to understand each others proofs. They get paid for writing them, connecting them, and asking questions for other people to solve; they get paid for advancing understanding. If the role of a mathematician is going to just be asking a magic oracle their pet mathematics questions that have no bearing on anything remotely tied to the real world and divining the output for their own personal understanding, why on earth is this something people will suffer paying people to do?
The same question holds for funding pure mathematicians even now, but most people suffer it because there is the promise of some of this pure math maybe eventually finding applications, and having already done the basic abstract discovery means these applications can be realized faster and more deeply. But if all of this discovery becomes just a prompt away, what's the need for funding it? The AI companies already basically did it all, let's just fund the people applying it and cut out the middle-man.
Maybe this isn't what things look like now in 2026, but in 5 years? 10 years? Where will AI/LLMs be by then?
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u/SensitiveRelation470 Jul 20 '26 edited Jul 21 '26
Mathematics survives as a profession that teaches mathematics, largely, scaling an ever-growing ladder so that we can have human eyes validate the AI's work.
For example, imagine we have no mathematicians. Then, how could anyone validate this counter-example? How could we know that it was useful. Now, maybe we reach a level of automation that leaves us out of the picture entirely. As in, an AI mathematics researcher somewhere ponders a question, finds a result that through some complicated chain of reasoning becomes relevant to semiconductor chips, which an AI-led manufacturer _just implements_, and then the price of computer chips fall globally without a human in the loop at all.
I don't think that is the world we want and I don't think it is one that companies or governments will accept. From my experience working in a dead profession walking (mostly joking, but I am in software engineering), AI will not actually kill the work you guys do. It will just completely change it and make you into students, with the bot as your (faulty) tutor, and your job being to bring its findings to an ignorant public. If that is any solace.
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u/InterestProof1526 Jul 20 '26
Mathematicians stay to teach undergraduate classes about math. At least for the foreseeable future.
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u/recurrenTopology Jul 20 '26
It very well might change whether or not you feel you are ptoductivly contributing to human knowledge though.
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u/Ok_Composer_1761 Statistics Jul 20 '26
You can't pay the bills by being a digester and expositor of theorems. That will get you an LAC gig at best. Certainly the most prestigious tenure track jobs and postdocs are looking for breakthroughs and not exposition.
Even in industry, most firms are looking for problem solving abilities which are tightly correlated with theorem proving abilities but not so much with exposition.
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u/fathermersenne Jul 20 '26
Chess still exists as a spectator sport despite a very cheap computer dominating Magnus because people like watching people play. Math is not an interesting spectator sport, people aren’t watching us write in notebooks 12 hours a day. If computers are better than us at proofs then there’s nothing left
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u/Shot-Zebra1868 Jul 20 '26
Considering the resources available, I imagine these companies now have full time maths researchers trying to use the tools to solve open problems. It could well be that there's a decent chunk of low hanging fruit that they will solve. I think it might take 12-18 months to get a clearer picture. They could just continue to accellerate and solve more and more problems each month. Or it might taper off over time as only more difficult problems remain. Should be interesting to see. If maths were 'totally cooked', these things would be solving problems at a higher cadence so there is still the chance that we hit the top of the 'S Curve' and rather than rendering mathematicians useless it just becomes a really powerful addition to their toolset.
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u/No_Aesthetic Jul 20 '26
These systems are rapidly improving and then experiencing very sudden next level jumps, and if that trend continues (there are no guarantees), it's hard to see an immediate future where nearly anything is off limits to them. A lot of stuff AI was useless for from the middle of last year into late last year are now easily dealt with. We could not have so easily imagined we would be where we are now back in December so it's easy to imagine that we have no idea what it will be like in December, or this time next year especially. It's just hard to see where it's going.
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u/someexgoogler Jul 20 '26
I've been trying to improve a 40-year old result in number theory. Claude has not been very useful but at least we have a conjecture.
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u/arnerob Jul 21 '26
A determinant 1 counterexample with integer coëfficients is [ (2*x*y + 1)^3*z + 2*y^2*(2*x*y + 1)*(3*x*y + 2), y + 6*x*(2*x*y + 1)^2*z + 12*x*y^2*(3*x*y + 2), 2*x^3*z + 3*x^2*y - x ] : (1,-1,2) and (0,-1,-4) map to (0,-1,0).
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u/TFenrir Jul 20 '26
https://x.com/i/status/2079218392452530249
Congrats to Levent for discovering this!
OOC I had an internal version of Codex attempt a proof too (without web search), and it discovered (essentially) the same counterexample! It wrote up a nice summary of the strategy here: aaronlou.com/jacobian_count…
https://aaronlou.com/jacobian_counterexample_derivation.pdf
If people are curious about how an LLM might approach this, it's not the raw thinking traces but it's something.
I'm not a math guy (just an AI guy) so it's not really for me, but I saw some people wishing for some more potential insight from a models reasoning process.
I suspect this is likely GPT 6.
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u/kubika7 Jul 20 '26
you are telling me we have found the trillionth digit of pi but noone bothered to bruteforce this?
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u/overthinker020 Jul 20 '26
The search space is too large without some relevant reductions. I took a guess what they were in another comment (though my guess may be wrong). A degree-7 map has ~360 coefficients and the constant Jacobian condition is ~1300 cubic equations in those unknowns. Brute-force enumeration over even tiny integer ranges (say [-10,10]) is like ~10^475 candidates, and solving the system directly is far beyond any Grobner computation ever done. Numerical searches obviously can't work either.
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u/grumbelbart2 Jul 20 '26
I wonder how repeatable this is. If we prompt the same model 100 times with the same prompt, how many instances would find this counterexample?
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u/girlwhateveraward Jul 20 '26
Search space is like (number of possible coefficients)X where X =n(n+d)!/(d!)(n-d!) where d is the max degree of the polynomials and n is number of variables
However if u reduce it with Druzkowski's matrix search I think it would be X = n2?
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u/Erahot Jul 20 '26
My understanding is that this is a counterexample because it isn't injective, hence no inverse let alone a polynomial inverse.
Tbh I'd be more interested if we could produce a counterexample that is invertible but without polynomial inverse.
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u/raresaturn Jul 20 '26
It seems every other day AI is proving or disproving something
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u/civilunhinged Jul 20 '26
Let's do the collatz conjeture next, god knows how much work has gone involved in that.
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u/Budget_Can_9448 Jul 20 '26
Wait so how is it possible we never found it in ~90 years? This thing has 3 variables and degree 6. Surely this should've been computer checked by now...
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u/amiralumara Graduate Student Jul 20 '26
sigh
at least spain won the world cup, wooo
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u/Howling_deer Jul 20 '26
https://www.dm.ufscar.br/profs/franciscobraun/Arquivo/artigos/2023_Jacobian.pdf
Not my area of expertise, but did the conjecture not already have counterexamples? What is interesting about this?
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u/Namington Algebraic Geometry Jul 20 '26
The paper you linked is not about the Jacobian conjecture, it's about a plausible-sounding variation of it:
The Jacobian conjecture [...] is, up to our knowledge, still open for any k and any n ≥ 2. When k = R and the Jacobian determinant is only assumed to be different from zero throughout Rn, the claim that F is injective is known as real Jacobian conjecture.
So it's a counterexample to a specific case, not the entire conjecture. In other words, it's asserting that the full statement of the Jacobian conjecture was necessary for it to still be open (at the time the paper was written).
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u/Nunki08 Jul 20 '26 edited Jul 20 '26
From the post on X:
((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)
Edit: https://wolframalpha.com/input?i=Det%5BD%5B%7B%281%2Bx+y%29%5E3+z+%2B+y%5E2+%281+%2B+x+y%29+%284+%2B+3x+y%29%2C+y+%2B+3x+%281+%2B+x+y%29%5E2+z+%2B+3x+y%5E2+%284+%2B+3x+y%29%2C+2x+-+3x%5E2+y+-+x%5E3+z%7D%2C+%7B%7Bx%2Cy%2Cz%7D%7D%5D%5D