A function of one variable can be translated in space by a spatial shift to obtain a new function

and also modulated in frequency by a frequency shift to obtain a new function

One can compose these two operations to obtain a time-frequency shift:

As per the time-frequency uncertainty principle, the two shifts do not quite commute with each other. For instance, we have

As such, is not a representation of the abelian group , but rather a portion of the Weyl representation of the Heisenberg group, but we will not adopt a representation-theoretic perspective here.

Some functions obey finite linear relations between their time-frequency shifts. For instance, a sinusoid obeys the relation

However, the Heil-Ramanathan-Topiwala (HRT) conjecture states that once one imposes some reasonable decay condition on , no such relations exist:

Conjecture 1 (HRT conjecture) If is non-zero, then there is no relation of the form

for some distinct time-frequency shifts and some coefficients , not all zero.

A special case of the HRT conjecture, which was also open, makes the additional assumption that was Schwartz.

Many positive results towards this conjecture were known. I will mention only a few here. A simple case is when we only have frequency shifts rather than spatial shifts:

In this case, the operator is simply a physical space multiplier

with symbol

so that the relation (2) now takes the simple pointwise form

If the coefficients are not all zero, then is a non-zero analytic function and thus has isolated zeroes. It is thus not possible to solve this equation for any . Thus the HRT conjecture is true when all the time-frequency shifts lie on the vertical axis. Using the metaplectic representation, one can then handle the case when all the are collinear.

What about the non-collinear case? Suppose first that all the lie in the lattice , thus for some integers . Here, the phase shift in (1) disappears, and all the time-frequency shifts commute with each other. This suggests that it should be possible to diagonalize the situation with a suitable transform to convert (2) to a pointwise equation similar to (3). To find this diagonalization, observe that if one restricts the function to a coset of the integers, then just multiplies the function by the scalar , while shifts the function on this coset by . The latter translation operation can also be converted to pointwise multiplication by performing the Fourier transform on the integers. Thus, if one introduces the Zak transform

of then the equation

can be transformed after a brief calculation to the equation

where the symbol is now given by the formula

As before, if the coefficients are not all zero, then is a non-zero analytic function and thus non-zero almost everywhere. Thus has to vanish almost everywhere, which for can be used to show that also vanishes.

More generally, there is a result of Linnell that the conjecture is true if lie in a translate of a discrete subgroup of ; this (together with the argument handling the collinear case) establishes all cases where , and several partial results involving the cases are also known. The conjecture is also known if is decays at a suitably super-exponential rate, by work of Bownik and Speegle.

I was aware of this conjecture through various talks and conversations with colleagues, and even briefly tried my hand at it for a while, though not with particularly serious effort (or progress). It was thus a nice surprise to see that it has just been resolved by Faulhuber, Petersen, van Velthoven, and Voigtlaender, even in the Schwartz case:

Theorem 2 There exist complex numbers , not all zero, distinct points , and a non-zero Schwartz function such that

It is perhaps unsurprising that this result is AI-assisted. However, I think the authors have disclosed their AI use responsibly, with the final arguments written by hand with a readable overview of the argument, as well as proper discussion of methods, relation to past literature, and other independent numerical checks on the result.

The negative result lies only a little beyond the positive results: is now increased to , and all but one of the points lie in (a translate of) a discrete subgroup of (in fact the explicit subgroup is used). The functions constructed are smooth and rapidly decaying, but not analytic or super-exponentially decaying, which would start being in conflict with the known positive results.

In addition to AI being used to come up with the initial proof strategy, a more traditional numerical computation was used to verify one step of the argument.

I have not had the time to do a full digestion of the result, but (after reading the introduction, and using a little AI assistance of my own) I was able to understand the main ideas at a high level. The first few reductions are relatively standard. Setting and , one can view the problem as one of solving an eigenvalue problem

The time-frequency shifts are chosen to lie in a translate of the discrete subgroup by a certain irrational shift . As mentioned previously, if in the shifts of a standard lattice , it would be natural to work with the Zak transform of , but it turns out that the approach does not quite work when doing this for topological reasons (relating to the fact that scalar quasiperiodic functions of mean zero are forced to have zeroes), and so the authors used the slightly denser lattice instead , which relates to a vector-valued version of the Zak transform taking values in rather than . Here, the phase shift in (1) does not completely disappear, but becomes a sign change. This slight loss of abelianness means that we cannot hope to diagonalize the problem all the way to a scalar problem, but we can still hope to reduce it to a two-dimensional vector-valued problem. Indeed, by applying a suitable vector-valued version of the Zak transform, the eigenvalue problem can be transformed to a a “vector cocycle problem”

where is a non-zero smooth quasiperiodic vector-valued function, is an irrational shift , and is a certain explicit matrix-valued function depending on the choices of , , and .

How to solve this equation? The motivating scenario here is if the matrix function was replaced by a rank one function

for some smooth vector-valued function of unit magnitude. Then one could solve the equation by taking and . It is not possible to make the function exactly of this form, but through some numerical computation and clever AI-assisted guesswork, the authors were able to find a choice of and , and that made approximately equal to a rank one function of this form, in fact getting a uniform estimate

As it turns out, such an approximation is sufficient to run a contraction mapping argument to find a solution to a variant of (4), namely

for some smooth and . (Here it was important to get the operator norm bound below ; they are barely able to do this, with a numerically obtained bound of , though this bound might not be optimal.)

The main remaining obstacle is that the “eigenvalue function” is varying in the parameter rather than constant. (This issue was, by the way, anticipated to some extent in previous work of Demeter, who observed that eigenfunctions of the almost Matthieu discrete Schrödinger operator gave a near-miss counterexample to the HRT conjecture, but with an eigenvalue that depended on an auxiliary phase shift parameter rather than constant.) However, if one was able to solve the scalar cocycle equation

for some smooth , then one could solve the equation (4) by setting . The approach to solve (5) is standard: take logarithms, apply a Fourier transform, and then divide out by the multiplier associated to the shift. This can cause a well-known “small divisor” problem (which arises in various dynamical contexts, such as in the KAM theorem) if behaves too much like a rational vector, but the standard resolution to this is to select a shift that obeys good Diophantine approximation properties. For the purposes of numerics the authors selected an extremely concrete shift, namely

but I get the impression that the exact choice here was not crucial for the argument, and that many other irrational algebraic numbers could have worked here.

Last September I went to a workshop at the Lorentz Centre in Leiden to discuss mathematics and AI with historians, philosophers, computer scientists, AI researchers, and mathematicians of several different flavours (though there was a surprising preponderance of algebraic geometers). The whole event was extremely stimulating, with some talks but also a lot of time set aside for discussion. One of the concrete outcomes of the workshop was the Leiden Declaration, which has now been signed by over 3000 people. Given that I was part of the workshop, it might seem a bit strange that I am not one of the signatories of the resulting declaration. The reason is not so much that I disagree with it in any concrete way, but more that in several places it makes confident assertions and recommendations that I feel somewhat uncertain about. So instead I prefer to try to articulate my views about the issues raised by the declaration and put them in this blog post. Before I do that, I would like to make clear that I am very glad that the Leiden Declaration exists and I think that it has done a lot of good in focusing people’s minds on the issues that AI is forcing the mathematical community to grapple with, which are more acute now than they were last September.

Let me begin by quoting a passage from the declaration that sets out “what we take to be characteristic values of mathematical research that we have a joint interest in preserving”.

1. There are many reasons to pursue mathematical research, ranging from intellectual curiosity to a desire to solve practical and societal problems. Underlying much of mathematics is the activity of proof. Mathematical proofs are regarded as conferring the highest degree of certainty to their conclusions, as well as imparting understanding of why their conclusions are true. These characteristics of proof support the scientific integrity of mathematics.
2. Results are attributable to specific authors who take credit for their discovery and assume responsibility for their correctness. These principles ground the merit-based standards to which we aspire in mathematical research.
3. Mathematical arguments are regarded as transparent and subject to independent verification. They may be extremely long or difficult, but in principle no proprietary knowledge or equipment should be required to understand them.
4. Mathematicians share a concern for proper evaluation of mathematical work relative to shared standards of depth, difficulty, and significance.
5. Mathematics produces not only a body of results, but also understanding, clarity, and judgment among the communities of mathematicians who have shaped them, often in the context of their own autonomously guided research. This expert knowledge is essential, both to effectively use mathematics, and to continue to articulate new and significant research questions. A key source of strength of the discipline has long been the autonomous shaping of the direction of research and the methods used to pursue it.

The first thing I would say about these values is that they are undoubtedly values that are widely held by mathematicians, including, with some qualifications, me. The main qualification I have concerns point 4: I find the notion of “proper evaluation” somewhat problematic, given that different mathematicians can have very different judgments without either of them being clearly wrong, especially when it comes to the significance of a piece of mathematics. Also, these judgments are used for purposes such as the acceptance of papers in journals, hiring and promotion decisions, the awarding of prizes, and so on, that are part of a system that copiously rewards a few people — I myself have hugely benefited from it — but doesn’t necessarily adequately reward a lot of people who are doing less visible work that is essential to keeping the whole enterprise going.

But the more important point is whether these values are ones that we should fight for in the future, as the Leiden Declaration suggests. I find that clearer for some of them than others. For example, it seems to me that the importance of rigorous proof will be even greater in an AI age than it was before — if the output of AI is not underpinned by rigorous proof, then the kinds of difficulties one already hears about with certain areas of human mathematics (see for example many talks by Kevin Buzzard arguing for the value of formalization) would be hugely magnified. But what about the attribution of results to specific authors, who take both credit and responsibility for them? Suppose that at some point in the future AI becomes more autonomous, reading the literature and solving many problems that it finds. Suppose also that its solutions are autoformalized, so there is no serious doubt about their correctness. In such a situation, there would be nothing for a human to take credit for or responsibility for. Does that mean that we should declare such results undesirable and threatening to mathematical values?

Of course, something could well be missing in such a situation: perhaps the proofs would be badly written and hard to follow, which would mean that they lacked something we all very much value. So let me extend the thought experiment slightly. What if by that stage one could take one of these outputs and ask an LLM to explain the ideas, and what if LLMs did a very good job at that? That is not particularly hypothetical, since they are often pretty good at this job already, but I am imagining a world in which they are much better than they are now, as they will presumably become.

So now we would have a world in which a lot of problems had been solved, we were sure that the solutions were correct, and we had an LLM ready to explain those solutions in as much or as little detail as we wanted. Is that a future we should resist, and if so, why?

One obvious reason is that it would take a huge part of the fun out of the subject. It is extremely satisfying to struggle with a mathematical problem for months or even years and eventually solve it. But I worry about that argument, because it seems to be saying that we should resist doing mathematics the easy way because a tiny fraction of the world’s population gets huge pleasure from taking orders of magnitude longer to do it. That is not to say that I wouldn’t be sad that a way of life that has sustained me for the last forty years was not available any more — of course I would. I just find it hard to use it as a reason to argue that we should try to preserve the “ownership structure” of mathematical results. If we arrive at a world where mathematical theorems are no longer associated with mathematicians, maybe that won’t be any more problematic than the fact that stars aren’t named after astronomers and most aren’t named at all. I’m not necessarily in a hurry for that world to exist, but maybe once the transition had happened, people would be OK with it.

The third value I share in an uncomplicated way, and I have already discussed the fourth. The fifth value is one that I hold very strongly, though I’m not so keen on the idea of experts consciously “shaping the direction of research”, something that I see as happening more organically. Obviously there are some notable examples of mathematicians who have created wonderful programmes of research, but even there I would like to credit other mathematicians with understanding what is wonderful about those programmes and contributing to them enthusiastically as a result, rather than being told what direction to pursue and meekly doing so (which is probably not what the declaration is actually trying to suggest, but it has a slight flavour of that for me).

But that’s a minor quibble when set against my main worry about the effect of AI on mathematics, which is the possible destruction of mathematical culture. There is at the moment an extraordinary body of knowledge and expertise that exists not just in the mathematical literature but in the heads of mathematicians all round the world. Imagine if AI didn’t exist and a pandemic broke out that for some reason wiped out all mathematicians and nobody else. All the literature would still be there, but nobody would have the faintest idea what to do with it. To revive a mathematical tradition under those circumstances would be extremely difficult and take decades. Now imagine a slight variant of that, where AI does exist and because of it people are no longer motivated to put in the years of effort it takes to reach the level of expertise that a typical research mathematician has now. After a decade or two, we might arrive at a situation where the mathematical literature has, in some form, been vastly expanded, but there is no corresponding community of human experts who have a shared understanding of parts of it. Almost all of mathematics would be like the areas that we have more or less forgotten about today, areas that exist in papers written many decades ago that nobody reads any more. (I won’t name any such area because I don’t want accidentally to suggest an area that many people still love and work on.)

This, it seems to me, is a possibility that we should try very hard to resist, but I agree with many other commentators who say that in order to resist it, we will need to give less priority to some of our current values — and I would include ownership of mathematical results in that list — and more to others. For example, if Person A gets an LLM to one-shot a solution of an important open problem (which is formalized, possibly automatically, so there is no doubt about its correctness) but Person B makes the effort to digest the solution and explain it in a way that other mathematicians can understand and learn from, then I think we will want Person B to get the lion’s share of the credit. The credit would be of a slightly different from what it is now, which could be described as admiration for somebody’s talent, insight, speed (I mean here the purely factual statement that speed is often admired — I would prefer that to be less the case) and hard work. It would be more like the gratitude that one feels already for somebody who writes a beautiful textbook that makes a whole area of mathematics coherent and accessible.

Maybe that is what the “research mathematicians” of the future should do: make a selection from a vast sea of AI-generated mathematics and write a book about it in such a way that other mathematicians can read the book and feel the kind of enrichment that we feel when we get to grips with an area of mathematics.

At this point I have to admit that there’s a pessimistic side of me that asks the following general question whenever anyone says anything about what the role for humans might be in the future: why do you think that AI wouldn’t be able to do it? For example, with the suggestion I’ve just made, what reason is there to suppose that ChatGPT 8.2 wouldn’t be able to have a short interaction with you about your mathematical tastes and background and then write the ideal textbook just for you? Humans are likely to be better at this kind of curating for a little while yet, but is it a fundamentally human ability that AI could never hope to emulate?

In a world where AI wrote bespoke textbooks (or more likely, just taught people in some more direct way), something would be lost that feels important: mathematics as a collective endeavour. If we all just learnt cool bits of maths for our own private satisfaction, we would miss the considerable pleasure that comes from discussing mathematics with others, though even that could in principle be restored by a benign LLM that deliberately taught many people the same cool bits of the subject, though an LLM that could do that sort of social engineering would raise all sorts of safety issues.

Let me now turn to the section of the declaration about potential threats. I’ll put my comments on each one in square brackets.

1. Current automated techniques can produce plausible but unreliable (or even incorrect) arguments which are difficult to distinguish from correct mathematical proofs. This applies not only to informal arguments, but also to formalizations, where the difficulty lies in the translation between computer-encoded and human presentations of concepts. These fast-moving developments put our present system of review under increasing pressure, jeopardizing our ability to implement traditional standards for the correctness, transparency, and independent verifiability of proof. [This feels like less of a problem now than it did last September, partly because the best LLMs hallucinate a lot less than before, and partly because autoformalization is improving all the time — I have just used harmonic.fun’s Aristotle system to formalize a complicated paper in Lean and I didn’t need to know any Lean to do it.]
2. Technologies that draw extensively on the published mathematical commons undermine the traditional system of attribution. Models trained on published works frequently return outputs that do not properly cite the human works they synthesize. Many current models are also built on data obtained by systematically exploiting licenses and access arrangements that were not made with artificial intelligence in mind, or indeed by simply violating copyright protections. [This is a problem at the moment, when ownership of results is important, and I am very much in favour of people making an effort to give appropriate credit for mathematical ideas that AI may have used. However, in the longer term, as I have already discussed, I think this ownership structure will break down and the issue will become less important. It also seems possible that LLMs will become better at revealing their sources.]
3. Technologies which affect the way in which mathematics is practiced may disturb the current system of incentives. The use of artificial intelligence — and thus also the sort of problems which it can address — may become incentivized for its own sake, disrupting our mechanisms for hiring, funding, and recognition. This disadvantages researchers who do not have access to the technologies or decision-making related to them, or who are unwilling to use technologies controlled by organizations whose values they do not share. [These seem to me to be genuine problems. I think there is simply no point in hoping that our current system of incentives will not be disturbed — it obviously will. I am not necessarily too worried if our mechanisms for hiring, funding and recognition are disrupted, as I don’t find those mechanisms unproblematic as they are, but disadvantaging researchers who do not have access to good LLMs is something I certainly think we should worry about.]
4. Proper evaluation is endangered if results are communicated through informal channels such as press releases or blog posts, often without any research paper or other disclosure of information necessary for scientific evaluation. This practice seeks publicity for new results on market timelines before the accepted processes of community evaluation in mathematics can take place. In many cases this leads to simplifications in reporting, such as overemphasizing the significance of automated tools and undervaluing the prior human contributions which have made those tools possible. Such oversimplification risks influencing public opinion in a way that not only damages perceptions of mathematics, but also misleadingly uses specific mathematical tasks as metrics for the general reasoning capacities of commercial products. [I think this can be a problem, but I think it is not as serious a problem as some of the others, since when results get overhyped, there seems to be no shortage of people publicly (and rightly) pointing that out.]
5. These developments put the autonomy of mathematics under threat. The increasing involvement of technology companies in mathematical research raises the risk that research questions may come to be prioritized because of their amenability to automated mathematics, rather than expert judgment of their deeper significance. Indeed, broader understanding of the field may be permanently lost in the process of automation. With university budgets under pressure, this reshaping also changes professional incentives in a manner which encourages the collaboration of researchers with technology companies on asymmetric terms. If left unchecked, these trends go beyond threatening researchers’ autonomy, affecting the scope and depth of mathematical research itself. [I think this could be a problem, but it also seems to me that mathematicians have a lot of power here. For instance, if a technology company were to produce a lot of research that mathematicians did not find all that interesting or important, I don’t think they would be able to use their financial and other resources to persuade us to change our minds. Rather, what seems to happen is that mathematicians say, “Yes that does X but it doesn’t do Y,” and the tech companies then feel challenged to do Y.]

There follow eleven recommendations for individual mathematicians. I agree with almost all of them. The one that I’m not so sure about, for reasons I’ve basically already gone into, is this.

Affirm the humanity of authorship. Credit and responsibility continue to belong to humans within the mathematical community and should not be given to automated systems. Artificial intelligence may obscure, but does not replace, the collective human labor behind a result.

I’m not sure what that really means. For example, should we affirm the humanity of authorship in the case of the solution to the unit-distance problem? Some humans did a wonderful job of explaining the proof that OpenAI’s model came up with, and the model made use of some highly non-trivial mathematics produced by humans, but the solution itself has not been credited to any human, and nor should it be in my view.

Under recommendations for mathematical organizations and not-for-profit research funders I again agree with several of them but have my doubts about some. An interesting case is the following.

Protect the rights of authors. Automated mathematics presents new challenges to the rights of authors, and societies should be proactive in the development of sample licensing agreements to protect these rights. In particular, material should not be used as training data without consent, and publishing agreements should allow authors to opt-out [sic] of the use of their work in this way.

This recommendation seems to belong to a world in which journal articles are the main means of dissemination of mathematics. But that has long since ceased to be the case: almost all dissemination now takes place via arXiv preprints, with journals limited to providing a little extra mark of prestige. Once an article is on arXiv, it is on the internet and one can hardly ask for it not to be used as training data. So this recommendation, if it applies at all, will apply to a tiny fraction of articles that are published without first appearing on arXiv. More generally, what right of an author is being compromised when an article is used as training data? We don’t object if human mathematicians use our articles to help train themselves to become better mathematicians — indeed, we will typically be delighted that somebody else thought our articles worthy of their attention. So the objection to a machine doing the same would have to be that for some reason one did not want machines to get better at mathematics in a similar way. I can imagine grounds for such a wish: perhaps somebody is worried about the threat that LLMs pose to traditional mathematical practice, or perhaps they worry that mathematical ability of LLMs will transfer to much more dangerous reasoning ability. But there’s a more complicated discussion to be had here than one might think from reading the recommendation.

The next recommendation is this.

Insist on appropriate publication outlets. Demand that mathematical results continue to be published in peer-reviewed venues such as journals, proceedings, and books. Informal mechanisms such as press releases or blog posts can provide a valuable supporting role, but they cannot replace peer-review or community scrutiny.

For reasons that I’ve gone into many times, I am not too fond of the current publication system, so I can’t get behind this recommendation. Indeed, if the current system becomes unsustainable because of a flood of AI-generated and AI-aided content, I would regard that as a beneficial consequence of AI. However, that doesn’t mean that I would advocate a total free-for-all. I’ve already said that one of my worries is that if mathematical content is not sufficiently organized, then the traditions that we all value could die. I just think that what we will want to do to preserve those traditions is likely to be a lot more innovative than clinging on to the peer-reviewed journal system.

I have highlighted in this post the parts of the declaration that I have doubts about, either because I disagree with them or, more typically, because I sort of half agree with them but want to add many qualifications. That may make the post come across as rather negative, but that is not my intention. The parts I disagree with are in the minority, and I think it is important that a declaration such as this should be made. I should also make clear that my views are evolving all the time, largely because the speed of progress of LLMs has taken me by surprise, but also as a result of conversations I have had or opinions that other mathematicians have expressed online.

I’ll end with two further clarifications. The first is that it may seem as though I am taking it for granted that LLMs will soon be better than humans at all aspects of mathematical problem solving, and maybe also problem posing, theory building, formulation of definitions, etc. I do think all that will happen at some point, but whereas some people say that it will obviously happen within the next two to three years, I would say that it might happen as soon as that, but I don’t rule out that we’ll get lucky and find that we can do interesting AI-assisted maths for quite a bit longer than that before AI doesn’t need us any more.

The second is that I think I have acquired a reputation as somebody who celebrates what is going on. But if, for example, I post on Twitter saying that such-and-such an AI solution is a remarkable development, the word “remarkable” is meant to indicate no more nor less than that I found it very surprising. My feelings about the possibility of AI solving all sorts of problems that interest me are much more mixed. I’ve had the experience twice now of seeing GPT 5.6 Pro one-shot a solution to a problem that I very much liked and had thought about hard (in both cases with much younger collaborators, who, with my approval, were the ones who prompted the LLM). It felt very strange and not particularly pleasant to have the rug pulled out from under my feet like that. On the other hand, I was quite pleased to see the problems solved. It’s actually a similar feeling to the one I have had many times when a problem I am fond of and have thought about gets solved by another human mathematician.

Another factor for me is that I have invested a lot of thought into automatic theorem proving of a more traditional kind. One of my main motivations for that was the hope that the work I put into it would extend the state of the art, measured by which problems a computer can solve. That ship has sailed now, and that saddens me. I still think that there is value in the work that I and my group are doing, but it has become a tougher sell.

So I personally have already found AI quite disruptive, and this is just the beginning. I would have preferred the developments to happen at a slower pace. But I don’t see any practical way to slow them down, so the best we can do is probably to face up to the changes that are being thrust upon us and do what we can to maximize the benefits and minimize the damage. The Leiden Declaration may not be perfect, but it makes an important and positive contribution to that effort.

The notorious Jacobian conjecture can be formulated concretely over the complex numbers as follows.

Conjecture 1 (Jacobian Conjecture) Let be a polynomial map in complex variables, whose Jacobian is a non-zero constant. Then is invertible (with polynomial inverse).

The condition that the Jacobian is non-zero is equivalent to being locally invertible. (The implication of local invertibility from non-vanishing Jacobian follows from the inverse function theorem; the converse implication can be derived from the Weierstrass preparation theorem, but is omitted here; see also Lemma 5 of this previous blog post.) Also, from the fundamental theorem of algebra, once the Jacobian polynomial is non-zero, it must be constant. So the hypothesis “Jacobian is a non-zero constant” can be replaced with “ is locally invertible”. So the Jacobian conjecture can be viewed as an assertion that local invertibility implies global invertibility. The complex numbers can be easily replaced with other fields of characteristic zero by the Lefschetz principle, but I prefer to work in the concrete setting of the complex numbers.

It was recently shown (using the Fable AI) that the conjecture is false in three dimensions (and thus in higher dimensions as well):

Theorem 2 (Counterexample to conjecture) There exists a polynomial which has non-zero constant Jacobian, but is not invertible.

The conjecture remains open in two dimensions, and is easy to establish in one dimension.

The example can be stated completely explicitly: one can take

and one can verify by a brief calculation that

and

While this is an extremely quick verification, the construction presented in this fashion appears like a massive miracle. The polynomial has degree seven, so a priori the Jacobian ought to be a polynomial in three variables of degree as large as , so the fact that all non-constant coefficients of this polynomial vanish looks like a massive cancellation involving equations, which is much larger than the degrees of freedom for a generic degree seven polynomial map of three variables. So finding such a polynomial looks highly unlikely to be located by brute force.

The example has since been retroactively explained in more geometric terms. As a “digestion” exercise to myself, I sought to write this explanation with relatively little use of algebraic geometry, in a manner that minimizes the amount of “miracles” required, although there are still a few places where some remarkable phenomena occur.

It is convenient to use the local injectivity formulation, and to generalize the domain to an equivalent affine variety. Namely, we will show

Theorem 3 (Counterexample, reformulated) There exists an affine variety that is isomorphic to by polynomial changes of variable, and a polynomial map which is locally injective, but not globally injective.

Clearly one can get from Theorem 3 to Theorem 2 by composing with the isomorphism and using the previously mentioned fact that local injectivity implies non-zero constant Jacobian. Our objective is now to find data , that obeys three separate properties:

• (a) is locally injective on .
• (b) is not globally injective on .
• (c) is isomorphic to by polynomial changes of variable.

The advantage of splitting the problem in to these three components is that we can build towards each of them separately.

(A pedantic remark: strictly speaking, in the arguments below, we not only replace the domain of by an equivalent variety , but also replace the range of by an equivalent variety . But the equivalence between and is a boring linear isomorphism ( will just be a hyperplane in a four-dimensional vector space ), so we do not highlight this aspect of the construction.)

It turns out that and can be built out of the operation of multiplication of low degree polynomials. Namely, consider the following three simple affine spaces:

• The space of linear homogeneous polynomials of two complex variables .
• The space of quadratic homogeneous polynomials of two complex variables .
• The space of cubic homogeneous polynomials of two complex variables .

(The notation here refers to the symmetric power of a vector space .) Clearly these spaces are isomorphic to respectively. Furthermore, we have a multiplication map , mapping a pair of a linear polynomial and a quadratic polynomial to a cubic polynomial

(Right now, the domain and range of this map is larger dimensional than the target of three; we will cut the dimensions down to three as the argument progresses.)

The map , essentially a map from to , is clearly polynomial; it is given explicitly in coordinates as

The map also enjoys two basic (and commuting) symmetries:

• If one applies a scaling for some non-zero complex numbers , then the product is scaled by : .
• If one applies a change of variables for some invertible linear transformation , then the product is transformed by : .

So this map enjoys a huge amount of equivariance, basically with respect to an action of the five-dimensional group .

The five-dimensional domain is of course larger than the four-dimensional range , so the map clearly cannot be injective. This can already be seen from the scaling symmetry, as the specific scalings

for modify the linear and quadratic polynomials but not their product . But even if one quotients out by this symmetry (3) to cut the dimension of the domain down to four, the map is still not injective for the following basic reason. A generically chosen cubic polynomial will split into the product of three independent linear polynomials. Then there are three pairs

which all map to the same cubic polynomial

under the multiplication map , but are not related to each other by scaling symmetry (3). Thus, we see that even after quotienting out by the scaling symmetry (3), the multiplication map is generically non-injective in a three-to-one fashion. Thus we already have achieved something resembling goal (b)!

It will be convenient to “spend” the scaling symmetry to obtain a useful normalization. If is a linear polynomial and is a quadratic polynomial, the (homogeneous) resultant can be defined by the determinant

If we have a factorization

then the resultant can also be described as

Thus the resultant measures whether the linear polynomial and the quadratic polynomial share a common root. A fundamental fact about resultants is that they are -invariant: for any , we have

One way to see this is to check it first for translations (which translate the roots by while leaving unchanged) and for inversions (which map to while mapping to and respectively), and then noting that these transformations generate all of . They also interact very nicely with scaling:

In particular, the scaling symmetry (3) multiplies by :

Thus, we can (generically) normalize away this scaling symmetry by imposing the condition

We now have a restricted multiplication map (which by abuse of notation we will continue to call ) from the four-dimensional variety

to the four-dimensional space . This map is still not globally injective, as we can take the three pairs in (4) from before and apply the scaling (3) separately to each of the three pairs to obtain the normalization (7). So we have kept property (b). Furthermore, this map retains the -equivariance (and also one remaining scaling symmetry, though we will not make much further use of that symmetry).

But we now also have property (a)! Suppose we want to show the local injectivity of in the neighborhood of a pair with . As the resultant is non-vanishing, the root of (which exists in the Riemann sphere, or projective line if you prefer) is distinct from the two roots of (though the latter two roots could be equal to each other). Applying the action (which performs Möbius transforms on the roots), one can assume without loss of generality that is the point at infinity (or equivalently ), thus for some complex number and for some complex numbers , with the resultant condition (7) simplifies to (so in particular are also non-zero). It is then clear that if one perturbs and by a small amount (say, modifying each coefficient by ), then the root of will perturb to something large (), while the roots of stay bounded. Thus, just from knowledge of the product , one can reconstruct which of the three roots of this cubic polynomial will be the perturbed root of , and which two will be the perturbed roots of ; from this and (6), (7) we can also reconstruct the leading coefficient of , and this completely determines both and . This establishes the local injectivity property (a). (In fact it is étale, but we will not need the machinery of étale maps here.)

Unfortunately, (the four-dimensional analogue of) condition (c) fails: the quadric hypersurface (8) is not isomorphic to the affine space . But we can try to get around this by passing to a three-dimensional slice. Let be some three-dimensional affine plane of (which we will take to avoid the origin for technical reasons), then we can restrict as a map from the set

to . is clearly identifiable (by linear changes of coordinate) to . As was already locally invertible, it remains locally invertible under restriction; and because generic cubic polynomials had three preimages under in (8), this continues to be the case after restricting to (9) (unless was somehow so degenerate that it had no generic elements, but this turns out to be impossible). So we have retained properties (a) and (b). The miracle is that, with a good choice of , we can also obtain (c) and obtain the desired counterexample to the Jacobian conjecture: despite appearances, the variety (9) is in fact equivalent to the affine space by polynomial changes of variable!

Let’s see how. The affine hyperplanes in avoiding the origin are parameterized by the dual space of avoiding the origin, which one can think of as the non-zero third order homogeneous differential operators in two variables. Indeed, every such operator generates an affine hyperplane that avoids the origin, and conversely by duality every affine hyperplane avoiding the origin arises in this form uniquely. Just as the cubic polynomials in can be factored into three linear polynomials, the differential operators in the dual space can also be factored into three linear differential operators, e.g.,

in the case that is non-zero. The action moves the roots around the Riemann sphere by Möbius transformations. As these transformations are -transitive, the actual selection of such roots is not too important (and the scaling symmetry similarly makes the choice of leading coefficient unimportant); the only thing to keep track of is whether the roots repeat. Up to the symmetries, there are in fact just three different equivalence classes of differential operator (and thus of affine hyperplane ) to consider:

• Operators where the three roots are all distinct, thus for independent first-order operators .
• Operators where two roots coincide and one is distinct, thus for independent first-order operators .
• Operators where all three roots coincide, thus for some first-order operator .

It turns out that the affine miracle for (9) occurs precisely in the second case, when has two identical roots. I do not have a completely satisfactory geometric explanation for this miracle, but one can verify it by the following coordinate computation.

By applying the action, we can normalize so that , thus is now the affine hyperplane of cubic polynomials with . Using (2) and (5), the variety (9) can now be described explicitly in coordinates as

At first glance this seems to be a generic-looking variety cut out by a cubic equation and a quadratic equation – hardly a candidate to be affine! But observe that if is non-zero, then the second equation can be solved for ,

and the first equation can be solved for ,

Putting these two equations together, we see that as long as one removes the case , the quintuple is uniquely determined by by a change of variables which is Laurent in and polynomial in . Thus we have a nice birational equivalence

Thus we have already almost established property (c): the variety (9) becomes birationally equivalent to after cutting out the subvariety. In particular, for each fixed non-zero value of , the corresponding fiber

of (10) is equivalent to by polynomial changes of variable, since we can reconstruct from the coordinates by the polynomial formulae

So we just need to glue back in the fiber. Indeed, from (10) we see that the fiber at is just

Now we observe a key miracle: the cubic equation and quadratic equation have a unique affine solution (as opposed to the six possible solutions that Bezout’s theorem might suggest – the other five solutions live on the line at infinity). So the fiber here is also affine:

This is extremely encouraging for the purposes of establishing property (c), as it strongly suggests that the variety (10) has the structure of an -bundle over , which is already extremely close to being isomorphic to the affine space . The main remaining task is to make sure that nothing singular happens in the limit , and that a global polynomial coordinate chart for (10) that covers both the and fibers can be constructed.

The standard way to proceed here is to manipulate various tangent spaces using the modern machinery of algebraic geometry and commutative algebra, but given my own background, I prefer to adopt the language of analysis, and in particular big-O notation (in place of the ideals used in algebraic geometry), in order to investigate the limit by hand. On the variety (10), let us use to denote any multiple of by a polynomial expression in . Thus, for instance, the equation implies that

while the equation implies that

as well as the more refined estimate

In the case we could conclude that . Now we perturb this observation. Multiplying (13) by we have , which on substitution into (14) gives ; substituting this back into either (13) or (14) also gives .

We can get some more precise asymptotics by also taking advantage of (15). Substituting into (15), we obtain after some algebra

So if we write more explicitly as , then we have

and thus

Substituting this back into (11) gives an asymptotic for :

Finally, one can insert these estimates into (12), although one only gets a trivial bound in this case:

Expanding the error term in (16) as , and doing a little more algebra, we thus have a polynomial change of variables

which completely parameterizes the variety (10) by polynomial combinations of three coordinates . This already gives (c) and thus completes the proof of Theorem 3.

The previous computations, when expanded out, also gives polynomial inverse maps:

The map from to the coefficients of (dropping the coefficient which is constrained to equal ), we obtain a polynomial map

with

which theory predicts to have a constant Jacobian, and indeed one can calculate that the Jacobian is . This is essentially the original example up to trivial changes of variable; indeed, one can check that the map

is exactly the map given in (1).

AI disclosure: I used an AI chatbot to discuss various aspects of this problem and to confirm several of the calculations made here.

Theorem of the Day has 'new acquisitions': Theorems no. 247-248

Theorem of the Day has 'new acquisitions': Theorems no. 249-251

Theorem of the Day has 'new acquisitions': Theorem no. 228, 229, 230, by R.A. Fisher, Poncelet and Ore

Theorem of the Day has 'new acquisitions': Theorem no. 231, 232, 233, 234

Theorem of the Day has 'new acquisitions': Theorem no. 235, 236, 237

Theorem of the Day has 'new acquisitions': Theorem no. 238 and 239

Theorem of the Day has 'new acquisitions': Theorem no. 240 and 241

Theorem of the Day has a 'new acquisition': Theorem no. 242, the Polya-Redfield Enumeration Theorem

Theorem of the Day has 'new acquisitions': Theorems no. 243-246

Theorem of the Day has a 'new acquisitions': Theorem no. 227, Cauchy's Theorem in Group Theory

Theorem of the Day has a 'new acquisitions': Theorem no. 226, Wolstenholme's Theorem

Theorem of the Day has 'new acquisitions': Theorem no. 223, 224, 225, by Tutte, Green and Regiomontanus

Theorem of the Day has a 'new acquisitions': Theorem no. 222, Faulhaber's Formula

Theorem of the Day has a 'new acquisitions': Theorem no. 221, the Inclusion-Exclusion Principle

Theorem of the Day has 'new acquisitions': Theorem no. 218, Riemann Rearrangement Theorem no. 219, Integration by Parts, and Theorem no. 220, the Pappus-Guldin Theorems

Theorem of the Day has 'new acquisitions': Theorem no. 216, Irrationality of Unit Circle Circumference and Theorem no. 217, Taylor's Theorem

Theorem of the Day has a 'new acquisition': Theorem no. 215, Wedderburn's Little Theorem