A counterexample to the Jacobian conjecture was published on a certain social media platform two days ago. It looks like a large language model was involved in finding this counterexample. Here's my random collection of thoughts about it.
The counterexample unfolded in two waves. The first message explicitly listed three polynomials and three points having the same image under the defined map. Anyone with the basic knowledge of algebra (say, a reasonable schoolchild) can verify two things: that the Jacobian of this map is constant (terms of pretty high degree miraculously cancel each other), and that the images of three given points coincide. The original message stated that the polynomials were found with the help of Fable (without specifying the degree of model's involvement).
The second wave came a couple of hours later, when several people shared transcripts of various chatbots that provided some higher-level explanation for the counterexample. It turns out, there's a geometric construction that is quite visual (if you are an algebraic geometer) and produces the given polynomials when written down in coordinates. See, for example, this writeup by Terence Tao that heavily relied on a chatbot. The geometric construction is quite simple and could (should?) be included in an introductory textbook on algebraic geometry.
So, what are we to make of this? It looks quite bad for human mathematicians: first of all, they `missed' a counterexample with polynomials that look quite reasonable. One can object to this that their degree is quite large (7 in the first found counterexample), and a naïve computer search for a counterexample would take astronomically long. But, what's worse is that they also missed the geometric construction involving things not more complicated than binary cubic forms. Some people said `mathematicians were trying to prove the conjecture, they weren't expecting it to be false, so no one has really been looking for a counterexample'. But is that really the case?
About twelve years ago, we were discussing the Jacobian conjecture a lot. My teacher, Nikolai Vavilov, gave an introductory series of lectures on it (in Russian), and we organized several talks on seminars that went much deeper. I distinctly remember Vavilov’s motto at the time: “Гипотезу якобиана не нужно доказывать, ее нужно опровергать” (“The Jacobian Conjecture shouldn’t be proved; it should be refuted”). So, in that sense, we expected there to be a counterexample (at least in higher dimensions), and I am pretty sure that was the consensus among serious mathematicians.
Here we have to make another important distinction: the Jacobian conjecture in dimension 2, and the Jacobian conjecture in dimensions 3 and higher. The counterexample(s) so far only work in dimensions 3 and higher. I feel that the general consensus was (and is) that the Jacobian conjecture in dimension 2 is true. Maybe it's even wrong to group the two questions together, and they should have been two separate conjectures.
Nevertheless, mathematicians dedicated a lot of energy to studying the Jacobian conjecture and its various implications. Another teacher figure for me personally (at least he was 30 years ago), Alexei Kanel–Belov proved (together with Maxim Kontsevich) that the Jacobian conjecture is (stably) equivalent to the Dixmier conjecture on endomorphisms of a Weyl algebra. I am not sure if Alexei was really trying to prove it, or if he was just exploring its neighborhood.
There is something psychological going on here as well: in cases where a counterexample is not obviously visible, it is very tempting to believe that the mathematical world is nice and smooth and beautiful. The Jacobian conjecture sounds nice and cozy.
What about the use of chatbots? We don't really know how the original author of the counterexample used LLMs, but we know that he is a mathematician of quite high level. We know that the prompt couldn't have just been `find me a counterexample to the Jacobian conjecture'; probably it was a long back-and-forth, trying different approaches and ideas (regardless of who generated them). The `second wave' (that generated a conceptual explanation) seems more surprising; but then again, similar constructions appeared previously in a paper by A. G. Vitushkin (on rational maps, not polynomial). It is not hard to imagine how a chatbot that consumed all of human knowledge available online (without any regard for ethics or intellectual property law) could pattern match the terms from Vitushkin's paper with the polynomial counterexample.
Note that all (?) of the much-discussed uses of LLMs in mathematics so far consist of finding counterexamples. Typically, verifying that a counterexample is valid is kind of easy: as I indicated earlier, a schoolchild can do it for the Jacobian conjecture. One notable exception is Erdős's planar unit distance conjecture, where a counterexample was found by finding significant links between combinatorics and algebraic number theory, and verifying it required using non-trivial theorems of class field theory. So, personally I am more impressed by the use of LLM for the planar unit distance problem than for the Jacobian conjecture.
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