Talk:Smale's problems

Table

There should be a table created for this article similar to one created for Hilbert's problems. Sr13 06:49, 11 November 2006 (UTC)

This article appears to have been copied verbatim from MathWorld. 68.40.190.221 06:43, 6 December 2006 (UTC)

You're right, thanks for bringing this to our attention. Since the MathWorld article has been around since 2004 and our article was only created this year, it's clear that our article was copied from MathWorld. The article is duly deleted. For further reference, the URL from which the article was copied is http://mathworld.wolfram.com/SmalesProblems.html . -- Jitse Niesen (talk) 07:04, 6 December 2006 (UTC)

I now rewrote the article afresh. -- Jitse Niesen (talk) 08:39, 6 December 2006 (UTC)

14-th problem

The 14-th problem links to http://en.wikipedia.org/wiki/Lorenz_system. That is not a problem! Nor is the solution, a link to http://en.wikipedia.org/wiki/Interval_arithmetic, actually a solution. The person who wrote the entry probably knows what they are writing about, but an encyclopaedia should exist to inform those who don't. 86.166.161.114 (talk) 17:31, 15 March 2015 (UTC)

8th problem

What's the status of the 8th problem? It's not clear from the text that's there. Apocheir (talk) 03:59, 31 October 2021 (UTC)

I don't understand it either. Dintre (talk) 08:17, 4 April 2023 (UTC)

18th problem

This problem is not sufficiently explained. This edit added dubious claims of solving the problem relying on a pair of recent studies. At least one of the two does not appear to have been peer reviewed and neither have much evidence of replication or independent verification. Moozle14 (talk) 03:09, 29 June 2024 (UTC)

Agreed - the claims are really quite dubious... I'm not knowledgeable enough in the subject area to properly assess (hence my modest edit just now rather than ripping it out), but the sources cited have a bad 'smell' to them. At the very start of the "unlimited intelligence" citation, they define a neighborhood as:

> The neighborhood of an element x is defined to be “a set of parts as a practical working whole” that contains x

Such a definition is very vague for a math paper... And the journal it is published in (Heliyon) seems to have had quality-control issues in the past. AlliterativeAnchovies (talk) 06:20, 9 September 2026 (UTC)

16th problem

> The $2$-dimensional case remains open; specialist literature has long distinguished the evidential status of the $2$-variable conjecture from that of the general $n$-variable conjecture.

@Sky19841

This is irrelevant to the resolution status of the conjecture as given in Smale’s list of problems, which is what this article is about. I think this sentence should be removed and (more importantly) the problem marked as resolved.

Pinging @HaeB Noaht2 (talk) 02:09, 21 July 2026 (UTC)

I agree that a "full/stable Jacobian conjecture" was disproved by a counterexample in dimension $3$.

However, Smale did not formulate Problem 16 as "it holds for every $n$". He wrote:"Suppose $f:C^{n}\to C^{n}$ is a polynomial map...Then must $f$ be one-to-one?" Thus $n$ occurs as a dimension parameter. In some usage this means universally "for any $n$"; but the same wording also defines a family of conjectures $JC_{n}$ or "Jacobian conjecture in dimension $n$, see Van den Essen, Peretz–Châu–Campbell–Gutierrez, Hamada–Kato–Komiya, Belov–Kanel-Kontsevich, Wright, et al; the universal version is often called full/stable Jacobian conjecture or $JC_{n}$, see in addition to above Truong, Bäck, Moskowicz.

We cannot confirm which version Smale really referred to, but this type of problem was firstly introduced by Kraus in $2$D in 1884; although Keller generalized to $n$-dimensions, Moh and Abhyankar, who gave the name "Jacobian conjecture/problems" studied primarily on the bivariate case. According to van den Essen and Formanek, the evidence in dimension $2$ is "overwhelming" and there's an "enormous difference" between dimension $2$ and higher dimensions (almost no evidence supports the higher dimension case), and this makes the $2$ dimensional case especially popular in research.

Thus, the yellow partial classification is more informative and less misleading than an unqualified green "Resolved". Sky19841 (talk) 07:03, 21 July 2026 (UTC)

Usually, in mathematical context when you don't specify $n$ or don't made additional assumption it is understood in full generality by default. The litearture about $JC_{n}$ you are referencing have different authors, there is no reason to assume that Smale follows the same notation.

Yellow classification (name "partial" in code) is not used to design partially resolved problems - it is used to design problems for there are some results, but there is no consensus whether this can be accepted as a result or not - currently 8th and 18th problem have that status. The problems with partial solutions that cannot be accepted as solution of general problem (or even cause controversy) are marked red (name "no" in code) - currently: 6th, 10th, 12th and 13th. Thus using yellow color for a problem you think is not resolved is more misleading.

As a consensus to avoid redaction war I propose to accept the most direct interpretation of Smale article, but put the following reference for planar Jacobian conjecture with all previous references:

"Despite counterexamples to general conjecture, the special case $N=2$ remains open, with specialist literature distinguishing its status from the general conjecture."

As I agree with you that planar Jacobian conjecture is interesting open problem and is certainly worth mentioning. MathInvestigator (talk) 23:24, 14 August 2026 (UTC)

“Usually, in mathematical context when you don't specify

n or don't made additional assumption it is understood in full generality by default.” You jumped a bit fast in this “usually”. Many parameters in Smale’s open problems are without “for any”, and in fact he discussed each solved case for those parameters in details rather than saying “this problem is just open”. This strongly indicates that his problems are parameter-indexed (as every mathematician knows, assuming something holds for every parameter is too strong, and we care about cases separately). A typical example is the Diff Poincaré conjecture. People would never say it is disproved although it has counter examples in many dimensions. You think that it’s a consensus that this problem has been resolved, while no mathematicians that focused on this problem before this event agreed so, only Moh has said “I only focused on 2D case”. I have 100% confidence that if Smale is concerned about the recent events, he would not agree so when he knew about it. This is not the way mathematicians would do with conjectures. Sky19841 (talk) 00:48, 15 August 2026 (UTC)

As explained by Smale, most of the problems are collected by others’ opinions. If he really considered the universal version, it would be quite striking that he included it as one of “problems of the next century”, because at that time both Moh and van den Essen noticed that the 2D case and higher dimensional case are highly different and in higher dimensions “almost no evidence supports it”. Van den Essen himself also explained he didn’t believe the universal versions. Why would Smale value a conjecture that more than 60% of the audience believed it was false (in a conference attended by van den Essen) and even there were almost no recorded attempts to look for counterexamples, except for Vitushkin’s 2D work? After that, van den Essen’s work on this was mainly about deriving formulations that result from or lead to JC(n). Sky19841 (talk) 01:22, 15 August 2026 (UTC)

I disagree with your interpretation. The differential Poincare conjecture that you mentioned is a great example: it was disproved by Milnor's proof of existence of exotic spheres. Is an open area of research to investigate if for some specific dimensions or specific cases of differential structures it holds despite was disproven in general: yes - but it is a set of different problems, because you added additional assumption. Every researcher working in this area must state clearly with which dimension or additional assumptions of diferential structure he works. In this way most of disproven conjecture you can consider "open" problem and never consider them closed, because most of them have some special cases where they holds.

I've read Smale article and for the problems that are expressed positively ha don't use phrase "for every". He written it in a slightly less formal language such as "can a compoex polynomial", "can a smooth map" (in problem 11th) and if you say to mathematician ther default understanding is "can every complex polynomial" or "can every smooth map". The Smale just asked in original statement if having a "polynomial map with non-singular derivative, must it be one-to-one?" No additional assumptions about dimension or extra properties of polynomial map. No distinguishing between dimension 2 and higher dimensions. Interpreting this simple question as a call for a complete program to clarify in which special cases Jacobian Conjecture is true if it was disproven is a highly personal interpreation, probably not obvious for the rest of mathematicians. In this way, the very special cases of Jacobian Conjecture still can be true in dimension $N\geq 3$ if you take additional assumptions about regular maps, so why treat existing counterexamples as solution for $N\geq 3$?

Why Smale considered important conjecture thought about half of specialist working in topic as questionable? You're right that Smale in the original article cites Bass, Connel & Wright and van den Essen works as a "background" and "importance of problem" reference and this authors distinguished status of $N=2$ and $N\geq 3$ basing on already obtined partial results. However, the general Jacobian Conjecture (despite believed to be false) had very important status in fields of analytic and algebraic geometry, since it promised very simple criterion for a regular mapping being invertible, which would greatly simlify many known results and prove some new, in general case for any dimension. This was the reason why (unexpected) proof or (expected) counterexamle for conjecture was of great importance. Despite 60% of specialist audience believed it to be false for $N\geq 3$, nobody given counterexample for 60 years between proposal of general conjecture (1939) and putting it on Smale's list (1999). MathInvestigator (talk) 09:09, 15 August 2026 (UTC)

By the way, I didn't found any information about Smale's death, so you can try to contact him and ask whether he supports your interpretation of Jacobian Conjecture status. If three different users inverts your editions claiming controversial status of this problem status or try to alter them, please take into account that your interpretation of this problem status may be very non-standard. MathInvestigator (talk) 09:22, 15 August 2026 (UTC)

Yes, since Smale hasn't appeared in public for a while and didn't react to the status of this problem, I had a wrong impression on his status. I'll contact him for this issue, and let you know when I get a reply (I'm not sure if he cares about the recent events, since he's 96 years old now). Sky19841 (talk) 10:45, 15 August 2026 (UTC)

You think that the "diff Poincaré conjecture" has been disproved and hence "resolved"? And so do you also think that the "Generalized Poincaré conjecture" is also resolved? If so, I think you are challenging the whole world of major mathematicians. In fact, whenever mathematicians discuss the status about these problems formally, we always distinguish the cases and say "XX is disproved" only if the remaining cases are really "negligible", but dimension is usually not a negligible parameter, that's the sense of the category "partially resolved conjectures". You think that the Jacobian conjecture is resolved, this is in some sense debatable, and this impression comes from the fact that the initial people (like Abhyankar, who brought this problem to major mathematicians and focused on 2D case) mostly either died or retired, and many people forget why this problem is "notorious". I believe after looking at the history, the major mathematicians will realize the issue (and in fact, still no mathematicians would say "JC" is resolved). But for the Generalized Poincaré conjecture, you really made a mistake, this is not a good argument.

Sky19841 (talk) 11:17, 15 August 2026 (UTC)

I distinguish the question I would call a Diff Poincare conjecture "Is every differetial manifold of homotopy type of sphere diffeomorphic to sphere?" from the question "In which dimension Diff Poincare conjecture holds?". I think I would challenge nobody with basic knowlendge of differential geometry if I say that the first question is resolved, the second could be considered resolved as soon as we have answer for every possible dimension. But they are two fundamentally different questions, moreover the second one arised following negative answer to the first. The first one is a conjecture - it gives clear thesis to prove or disprove, the second one is a problem - it offers no clear thesis, maybe a broader topic to research. Not every open problem needs to be conjecture: the good example is Hilbert's 15th problem where Hilbert asked for strict formulation of Shubert calculus, with no proposed thesis to prove or disprove.

For generalized Poincare conjecture as presented here if treated as a conjecture for every category of manifolds it is obviously resolved, because fails for differential manifolds. The problem "for which category" is obviously unresolved, similarly with a problem "for which dimension" in categories where it could fail. I'm unsure if this problem could ever be considered fully resolved: Wikipedia article enlist three categories of manifolds of greatest importance, but I can imagine that mathematicians could give many more of them. However, you have good point here: the formulation of conjecture "for a category of manifolds" is so vague that probably every specialist would immadiately demand precise category you are using. But the difference is that we don't have well-defined object of "all categories of manifolds" but we have well defined set of natural numbers that reflects dimensions.

For the JC we may wait for specialist opinions, since only a month passed since counterexample. But there are some another famous conjectures that was disproven and this was widely accepted as a resolution: one example could be Hilbert 14th problem about finite generation of some algebras, the another one 21st about existence of Fuchsian equations. Of course, some researchers still work to construct counterexamples for rings of smaller dimension, but this is a follow-up, the original Hilbert question was resolved. For 21th the additional assumptions that guarantee existence are so weak, that they rather closed further research.

But I would give you another question in return - the Smale's list contain also Riemann Hypothesis and P = NP problem. If there would be found a zero of Riemann zeta function off critical line and Riemann hypothesis would be disproven - would you accept it as a resolution of conjecture? Or demand comlete theory about behavior of nontrivial zeros? If would be find problem in NP class that is clearly not in P - would you accept it as a resolution? Or demand complete computational theory that could distinguish every NP problem to be P or not P? By specialists in respective topics the first one is generally believed to be true, the second one to be false (by the way, another problem believed to be false, but any counterexample would be of great value) but I noted that our controversy concerns disproved conjectures. MathInvestigator (talk) 08:22, 16 August 2026 (UTC)

The problem is, you made a scheme of conventions on "conjectures", and think that they must uniformly follow some rules, like "the parameters are universal, and disproving one parameter implies disproving the whole conjecture"; while there's no such common rule in the broad world of mathematicians; some mathematicians choose the words rather casually, because we believe everyone in the field should understand the importance of the works, and what is true/false, although people from the outside may misunderstand. When mathematicians talk about "conjecture" with some parameters (without "for any"), it often happens that they in fact mean "a set of conjectures indexed by these parameters" rather than "a conjecture that holds for universally all parameters" (for JC, the case is more involved. Near the origin of this naming, some mathematicians like Abhyankar only referred to the 2D version). The convention usually depends on the importance of the separate cases, and differs among problems. I work on harmonic analysis, and one of the most famous problems in my field, the Kakeya conjecture is in fact dimension-indexed. Even if in future, it is disproved in some higher dimensions (although I suspect such possibility), no one in this field would say "the Kakeya conjecture is disproved". The generalized Poincaré conjecture is another typical example. I'm not sure which field you are working on, but none of my colleagues working on differential topology would say "'the Diff Poincaré conjecture' is disproved", because they primarily referred to it as a dimension-indexed conjecture, and in fact I'm quite astonished to see someone referring to it as the universal version "for all dimensions". It reminds me of the scene of learning English, when people arguing "this word cannot be used in this way", sometimes it makes sense, while sometimes it's just ridiculous and no one would care in real life. Someone even uses "conjecture" to refer to conditions (e.g. Miyanishi, who first in published record referred Jacobian conjecture to n-dimensional version, while he later had many works on "generalized Jacobian conjecture" but provided counter example for specific cases in the same works). The RH or (P versus NP) are not relevant here. They have no parameters to separate the cases and for sure are universal. Hilbert 14 and 21 are also not good examples here. First, Hilbert asked problems instead of making conjectures (even if he thought them as possibly true), Second, Hilbert 14 is solved in any cases now, but for mathematicians in this field, Nagata's work is the most important, this is based on their inside evaluation of the work (the remaining cases did not attract attention as very special cases in the literature). And people can definitely say "Nagata provided a counter example to the 14th Hilbert problem", "Nagata's solution", but whether or not it is "solved" at that time might be controversial, because, after that, there is a 7 dimensional counter example by Roberts, 6 dimensional counter example by Freudenburg, 5 dimensional counter example by Daigle–Freudenburg, 4 dimensional counter example by Kuroda, and finally 3 dimensional counter example again by Kuroda. These are still important works in this field. If the problem is regarded as "solved", these works would not be published in respected journals. People should be cautious about this wording, sometimes there is no absolute "solved" or "not solved", and the distinction mostly depends on importance of each work and separate cases. As for Hilbert's 21st problem, the expression by Hilbert himself is already ambiguous and controversial (e.g. he used "equations" instead of "systems", which had been disproved before, and he mixed Fuchsian with regular singularity), but for sure all these cases are either proved or disproved, and people don't regard it as open. Sky19841 (talk) 17:59, 16 August 2026 (UTC)

Ok, thanks for sharing Your mathematical background. I know barely nothing about harmonic analysis, so I will trust you that yours approach is standard in this field of mathematics. Let me share mine background: in a subfield of algebraic geometry which I current study is very strict about names of different theorems, conjectures or problems, especially when they are closely related. Treating dimension as a free parameter would be controversial, because I expect that the questions arise: "if you treat dimension in this way, why would you not treat assumptions about number field which this variety is defined over? Or divisors of variety? Or abelian structure on variety? Or Calabi-Yau structure on variety? ..."- and in this way original conjecture that was disproven could never be considered resolved.

This was also the reason why I asked you about another assumptions about regular mappings in higher dimensions or about additional assumptions for differential structures on spheres. You're right that journals won't publish uninteresting article that don't bring anything new into mathematics. But I'm pretty sure that in next years we can expect articles proving Jacobian conjecture for very special cases of maps in higher dimensions. Exactly: because original problem in this dimension have negative resolution, the positive follow-ups become more interesting.

For me, you proven well your point about classification of 16th problem. My background standards is a reason why I consider "original" Smale problem resolved and treat dim = 2 as different conjecture under extra assumption. But if standards accepted by two different branches of mathematics classifies current result differently, it is lack of general mathematical consensus and justifies yellow color. I don't plan to change status of problem unless a definitive reult for dim = 2 will show. (May be a few months or a few decades? - nobody knows for sure).

By the way, thanks for updating article about Hilbert's 14th problem! I greatly appreciate sharing knowlenge about that counterexamples with rest of readers. MathInvestigator (talk) 17:56, 23 August 2026 (UTC)

Yeah in reality, when the problem doesn’t include “for all parameters”, the standard answer from mathematicians on whether or not a problem is solved simply depends on how "beautiful" or important these cases are, regarded by the specialists. Borsuk's conjecture is an example on the other side. That conjecture is about values of a function $f(n)$ on dimensions. He conjectured it's $n+1$, while it was proved to grow exponentially, that makes his conjecture in some sense not that striking and regarded naturally as disproved. Although people will still try to find $f(n)$ for $n=4$, but this is just one problem, not something like “solving it helps to discover some unique structures”. For Hilbert 14, if one has to say it is solved by "xx", then it's definitely Nagata, because that is regarded as the most important (although I usually avoid saying so, since the affirmative result by Zariski still seems interesting and nontrivial). While for Jacobian, the structural difference is essential. If the 2D case is really true, then its beauty (while maybe sounds a bit subjective, but at least for many mathematicians in this field) and the extreme difficulty would make the people who prove it feel extremely ecstatic. That's the essential difference. It’s cold after 2000s mainly because people have used up known technics to deal with it, and every new idea they can think of is independent of dimensions (like, the equivalence formulations).Sky19841 (talk) 19:39, 23 August 2026 (UTC)

Jacobian brevity

As far as the Jacobian conjecture's status is concerned, shouldn't the list simply state that the found counterexample disproves the conjecture for ⁠$n>2$⁠, as it is formulated in Jacobian conjecture, rather than indicate $N=3$ and "counterexamples for ⁠$n\geq 4$⁠" separately as it stands now? Ur frnd (talk) 18:59, 9 September 2026 (UTC)