Talk:Jacobian conjecture

Untitled
Firstly, on the conjecture. The stronger form is that F can be inverted by polynomials: i.e. that F gives a ring automorphism of the polynomial ring. Is this known to be equivalent.

Secondly on Carolyn Dean: what I know from the Web:

- she wrote a PhD supervised by Lance Small at University of California San Diego in 1986

- she is now a lecturer at the University of Michigan

- she will be presenting the proof at talks starting December 2.

On the proof, Melvin Hochster is now vouching that it is correct. It depends on some earlier results in the field.

A few days ago I posted here that I thought the proof for n = 2 by Carolyn Dean had been withdrawn, and stated that I was in the process of verifying this.

Today (1/1/06) I have received an e-mail from Melvin Hochster confirming that, indeed, the proof was retracted soon after it was announced (aforesaid talks on Dec. 2, 2004 never took place), and that Carolyn Dean is still trying to repair the flaw in her proof.

So for the time being it would be best to remove this phrase in the article:

<< The Jacobian conjecture has been proved for polynomials of degree 2,

Sincerely,

Dan AsimovDaqu 19:20, 1 January 2006 (UTC)


 * But that is degree d = 2, not number of variables N. Charles Matthews 21:27, 1 January 2006 (UTC)

Thanks for pointing out my silly confusion, Charles. I was influenced by the statement at top of this page, which perhaps should be a) dated and b) updated. 4.239.30.27 02:03, 2 January 2006 (UTC)

There seems to be a feeling that for large N the conjecture may turn out to be false.

Charles Matthews 15:11, 21 Nov 2004 (UTC)

Two web references for the story:


 * http://groups.google.com/groups?dq=&hl=en&lr=&group=sci.math.research&selm=d29cmre258j5%40legacy
 * http://www.matrix.ua.ac.be/index.php?p=58 Charles Matthews

origin of name
Moh's webpage, listed in the external links, explains he gave the conjecture this name. What's the reference for Abhyankar naming it? --C S (Talk) 06:08, 4 February 2007 (UTC)


 * More pertinently, why is it called "Jacobian" -- after which Jacob or Jacobi? Equinox ◑ 23:46, 29 July 2018 (UTC)


 * Ah, found the answer at Jacobian matrix and determinant. Equinox ◑ 23:51, 29 July 2018 (UTC)

Further reduction
Shouldn't the result from http://www.ams.org/journals/proc/2005-133-08/S0002-9939-05-07570-2/home.html be included in the Results section as well? — Preceding unsigned comment added by 131.174.91.213 (talk) 15:35, 28 February 2012 (UTC)


 * I just added this. Someone else may add the link as a source. I'm not quite sure how to do this and the overview of references seems quite inconsistent, when it comes to lay-out. HSNie 09:37, 16 September 2014 (UTC)

Clarification
I'm confused by the statement of the conjecture on this page. Can somebody help clarify? What does it actually mean to say that the matrix is a non-zero constant? Does that mean it's allowed that the matrix is singular, as long as it has atleast one entry that isn't zero? Is a constant matrix equivalent to a diagonal matrix, or a matrix where all entries are equal? — Preceding unsigned comment added by 67.164.143.201 (talk) 17:38, 6 October 2013 (UTC)

Re "invertibility of the Jacobian is equivalent to the Jacobian matrix being nilpotent". Surely that should be "non-invertibility"?

Improvements to JC article
Preface. This note is intended to suggest improvements to the Jacobian Conjecture article last modified on 4 May 2015.

I am a mathematician and have written a large number of journal articles on the JC. Most of my remarks will be just that and are intended as suggestions only when explicitly identified as such. I will not make any edits myself. Not only am I a newbie, but I am also blind and my text-to-speech program does not distinguish between upper and lower case letters. Whoever does the edits will have to deal with some minor notation issues.

JC Summary. Let k be a field of characteristic zero, n a positive integer, and F: k^n -> k^n a polynomial map. For fixed k and n, the JC is the assertion JC(n,k) that any F with a nonzero constant Jacobian determinant has a polynomial inverse. It is trivially true for n=1. It  has not been proved for  any (n,k) with n > 1. Many special cases have been examined and no counterexamples have been discovered. It is known that if JC(n,k) is true for any one k that is algebraically closed, then it is true for that n and any k. So for each n, it suffices to prove the JC for the complex field C. By separating complex values into their real and imaginary components, one can show that JC(2n,R), where R is the field of real numbers, implies JC(n,C). Thus the JC for all n is true for all k if it is true for R. Actually, it is known that if the JC is false, then there is a counter example with integer coefficients and Jacobian determinate 1. That is then a counter example for any k. Thus if JC(n,k) is true for all n and any specific k, then it is true for all n and all k.

These results, together with proofs, attributions, and historical notes, can all be found in Arno van den Essen’s great book on the JC – see below. He earlier wrote a survey paper with the same title as the book. The Wikipedia article references conflate the two.

Suggestion #1. Fix the references. Include the book for sure, as it is the authoritative reference for the JC in the 20th century.

The following citations were obtained by using MR Lookup (freely available for limited use).

van den Essen, Arno. Polynomial automorphisms and the Jacobian conjecture. Progress in Mathematics, 190. Birkhäuser Verlag, Basel, 2000. xviii+329 pp. ISBN: 3-7643-6350-9 MR1790619 (2001j:14082)

van den Essen, Arno. Polynomial automorphisms and the Jacobian conjecture. Algèbre non commutative, groupes quantiques et invariants (Reims, 1995), 55--81, Sémin. Congr., 2, Soc. Math. France, Paris, 1997. MR1601194 (99b:14015)

Suggestion #2. This concerns section 1 on the Jacobian determinant. The Jacobian matrix is not formally defined, even though the term is used later. Define it first, then the JD. I usually use J(F) for the JM and j(F) for the JD. Next, the conjecture is not about algebraically closed fields, since they may be of characteristic p > 0,  in which case the JC fails even for n=1 (x-x^p has constant derivative 1, but is not one-to-one). Just say 'in a field'. Don't introduce C, and use the letter a to name the components of a point, instead of c. I would recommend  stating that the JM is a matrix with polynomial entries and finishing off the section with the following: Using the multiplicative property of determinants and Cramer's rule, the JM has an inverse matrix with polynomial entries if, and only if, the JD is a nonzero constant.

Note. Inclusion of the last part of the suggestion is probably a matter of taste. But it is true for any field k.

The JC can be stated for commutative rings with multiplicative identity element 1, but it is natural to restrict the formulation to fields. One must assume the ground field is of characteristic zero, meaning that no finite sum 1+1+1 ,,, +1 is equal to zero. Any field of characteristic zero contains a subfield (generated by 1) isomorphic to the field of rational numbers Q. So it suffices to formulate the JC for extension fields of Q. Either way, all the fields are infinite, so that polynomial functions and polynomial maps are determined by their values at all points. It follows that if F has a polynomial inverse, then the inverse is unique both as a map of sets and as a polynomial map. A polynomial map may have an inverse that is not polynomial (x+x^3). If F is polynomial with polynomial inverse G and all coefficients appearing in F belong to a subfield, then that is true of G as well. For instance, if F has rational real number coefficients, then so does G. That is why JC(n,k) for an algebraically closed field k and a fixed n yields the JC for that n and all subfields of k. Results from mathematical logic (Model Theory)  and also some more direct arguments can be used to show that  any single algebraically closed field of characteristic zero, for instance C, is good enough. This is all background, not intended for insertion into the Wikipedia article. I hope it is now clear that both the proposition and formulation of the JC in section 2 contain mathematical errors.

Suggestion #2. The following is a draft substitute for section 2 on the formulation of the conjecture.

If F has an inverse function G: k^n- > k^n, then applying the multivariable chain rule to the equations g_i(f_1,,,,,f_n)=x_i shows that the Jacobian matrix of F has an inverse matrics with polynomial entries, and hence that jF is a non-zero constant.

The conjecture is the following partial converse:

Jacobian conjecture: If k is a field of characteristic zero and jF is a non-zero constant, then F has an inverse function G: k^n - > k^n, and G is regular (in the sense that its components are given by polynomial expressions).

Easy examples show that the restriction on k is necessary. It is satisfied by any field containing the field of rational numbers, for instance the real or complex numbers.

For real variables and functions the condition j(F) not = 0 is related to the inverse function theorem in multivariable calculus. In fact for smooth functions (and so in particular for polynomials) a local inverse function to F exists at any point where jF is nonzero. The Strong Real Jacobian Conjecture was that if F is a real polynomial map with a nowhere vanishing Jacobian determinant, then it has a smooth global inverse G.  Counterexamples for any  n>1 were found in 1994.

Suggestion #3. The following or similar remarks should be inserted either at the end of section 2 or the beginning of section 3.

The Jacobian conjecture is true for all fields of characteristic zero if it is true for a single specific field of characteristic zero, provided that all values of n > 1 are considered. Thus it suffices to prove it for, say, the real case. For a specific fixed value of n > 1, it suffices to prove the result for some algebraically closed field of characteristic zero.

Suggestion #4. Add this reference.

Pinchuk, Sergey. A counterexample to the strong real Jacobian conjecture. Math. Z. 217 (1994), no. 1, 1--4. MR1292168 (95g:14018).

Math. Z. is an abbreviation for the journal name Mathematische Zeitschrift. Probably the full name should be used.

L.Andrew Campbell (talk) 05:34, 23 June 2015 (UTC)

Jacobian conjecture results
I would like someone to edit the results to produce roughly the following surface appearance:

Wang (1980) proved the Jacobian conjecture for polynomials of degree 2, and Bass, Connell & Wright (1982) showed that the general case follows from the special case where the polynomials are of degree 3, more particularly, of the form F = (X1 + H1, ..., Xn + Hn), where each Hi is either zero or a homogeneous cubic. Drużkowski (1983) showed that one may further assume that the nonzero Hi are cubes of homogeneous linear forms. These reductions introduce additional variables and so are not available for fixed N. Connell and van den Dries (1983) proved that if the Jacobian conjecture is false, then it has a counterexample with integer coefficients and Jacobian determinant 1. In consequence, the Jacobian conjecture is true either for all fields of characteristic 0 or for none. For fixed N, it is true if it holds for at least one algebraically closed field of characteristic 0. Moh (1983) checked the conjecture for polynomials of degree at most 100 in two variables. deBondt and van den Essen (2005) and (Drużkowski) (2005)  showed that it is enough to prove the Jacobian conjecture in the case of complex maps with a symmetric  Jacobian matrix. The strong real Jacobian...

Besides the treatment of math symbols and citations, the name Drużkowski needs a dot over the letter z. The ... indicates a point after which there are no further changes to the result section. The first sentence is unchanged. I removed the statement about unipotence, because it is a technical detail not appropiate here. Note the change in spelling to deBondt( there should be no space). There are several new references shown below. Also note that the references are not currently in the usual alphabetic order.

I plan to make some further changes later, specifically to address the birational Galois cases of the conjecture.

Connell, E.; van den Dries, L. Injective polynomial maps and the Jacobian conjecture. J. Pure Appl. Algebra 28 (1983), no. 3, 235--239. MR0701351 (84m:13006)

Drużkowski, Ludwik M. An effective approach to Keller's Jacobian conjecture. Math. Ann. 264 (1983), no. 3, 303--313. MR0714105 (85b:14015a)

Drużkowski, Ludwik M. The Jacobian conjecture: symmetric reduction and solution in the symmetric cubic linear case. Ann. Polon. Math. 87 (2005), 83--92. MR2208537 (2007a:14068)

L.Andrew Campbell (talk) 20:08, 21 July 2015 (UTC)


 * I've added the missing diacritics to Drużkowski's name. I hope you don't mind.  You can copy and paste this into the article yourself, or I can do it for you if you prefer.  Best,  Sławomir Biały  (talk) 00:04, 22 July 2015 (UTC)


 * I've merged that into the article. The reference by deBondt and van Essen (2005) seems to be missing.   Sławomir Biały  (talk) 11:57, 26 July 2015 (UTC)


 * I was wrong about the spelling of de Bondt. Will fix in article. I agree the reference is missing.  Here it is

de Bondt, Michiel; van den Essen, Arno. A reduction of the Jacobian conjecture to the symmetric case. Proc. Amer. Math. Soc. 133 (2005), no. 8, 2201--2205 (electronic). MR2138860 (2006a:14107)

L.Andrew Campbell (talk) 18:30, 30 July 2015 (UTC)


 * Thanks, I've added the reference.  S ławomir  Biały  12:56, 31 July 2015 (UTC)

Mr. Campbell. First of all I must say I really appreciate the work you've done on this article so far. I've got one objection to make, though, after having worked with A. v.d. Essen and M. de Bondt for some time. As you've already found out, the spelling of the latter's name really includes a space. However, you removed the remark by Moh about nilpotency. If you'd talk to v.d. Essen, you might find out that he actually finds this a most important result. For that reason alone, I'd suggest reinstating it in the article.

In the same vain, the introduction now mentions an article by same person from 1997. In 1997, Arno didn't believe JC to be true. In the years following it, he changed his mind and considered JC to actually to be true (as he does now). Maybe using a reference from 1997 by this author in the introduction might not be a good idea. HSNie 00:13, 19 August 2015 (UTC) — Preceding unsigned comment added by HSNie (talk • contribs)

Abhyankar
In the article it says, Abhyankar  ‘named and widely publicized’  the conjecture. I will change that to ‘widely publicized’. I personally encountered the conjecture in grad school at Princeton in the late 60’s. It was called the Jacobian Conjecture. While Abhyankar used that name in the titles of his later papers, he appears to have consistently called it the Jacobian problem early on.

I haven’t read this entire reference but it is available on line:

Ghorpade, Sudhir R. Remembering Shreeram S. Abhyankar. With appendices by Avinash Sathaye, Balwant Singh, Steven Dale Cutkosky and Rajendra V. Gurjar. Asia Pac. Math. Newsl. 3 (2013), no. 1, 22--30. MR3060712

It appears as a Jacobian problem in: Abhyankar, S. S. Lectures on expansion techniques in algebraic geometry. Notes by Balwant Singh. Tata Institute of Fundamental Research Lectures on Mathematics and Physics, 57. Tata Institute of Fundamental Research, Bombay, 1977. iv+168 pp. MR0542446 (80m:14016)

and also in a chapter heading in:

Abhyankar, Shreeram S. Algebraic geometry for scientists and engineers. Mathematical Surveys and Monographs, 35. American Mathematical Society, Providence, RI, 1990. xiv+295 pp. ISBN: 0-8218-1535-0 MR1075991 (92a:14001)

L.Andrew Campbell (talk) 01:15, 6 August 2015 (UTC)

The birational and Galois cases
As usual, I will need help.

Both cases use concepts from modern algebra and so are closer to the formulation in abstract algebraic geometry. I have inserted a bare bones description of the basic facts. I have omitted mentioning that Abhyankar proved the two variable Galois case sometime in the early 70s, since he used methods not on the path to the general solution.

Before getting to it, let me mention some results that make it easier to understand. Since the JD is not the zero polynomial, the fi are algebraically independent. If each Xi is a polynomial in the fi, one gets a left inverse, say G, to F. That is, G o F = I. Then one can use algebraic independence of the fi both to show that G is unique and that F o G = I.

Suggested mod:

The following is the desired approximate surface appearance of the involved stretch of text. It begins and ends with an unchanged sentence. Also, delete the existing reference to Adjamagbo (1995) from the results section, but do not delete the  citation itself. The 3 new references needed follow the mod.

For fixed N, it is true if it holds for at least one algebraically closed field of characteristic 0. For a given F, it is true if k[X]=k[F] (the Xi and the fi generate the same k-algebra). Keller (1939) proved the birational case k(X)=k(F) (they generate the same field extension of k). The Galois case (k(X)/k(F) is a Galois extension) was proved by Campbell (1973) for complex maps and in general by Razar (1997) and, independently, Wright (1981). Adjamagbo (1995) suggested extending the Jacobian conjecture to characteristic p > 0 by adding the hypothesis that p does not divide the degree of the field extension k(X)/k(F). Moh (1983) checked the conjecture for polynomials of degree at most 100 in two variables.

Here are the references:

Campbell, L. Andrew. A condition for a polynomial map to be invertible. Math. Ann. 205 (1973), 243--248. MR0324062 (48 #2414)

Razar, Michael. Polynomial maps with constant Jacobian. Israel J. Math. 32 (1979), no. 2-3, 97--106. MR0531253 (80m:14009)

Wright, David. On the Jacobian conjecture. Illinois J. Math. 25 (1981), no. 3, 423--440. MR0620428 (83a:12032)

```` — Preceding unsigned comment added by L.Andrew Campbell (talk • contribs) 18:14, 13 August 2015 (UTC)

Suggest removal
The following text was recently inserted in the opening paragraph of the Jacobian Conjecture article:

What is perhaps the simplest unsolved case of the conjecture conveys its difficulty: Let f: C2 ? C2 be a polynomial mapping. (That is, f(z,w) is of the form (P(z,w), Q(z,w)) where P and Q are polynomials with complex coefficients). Assume that f is locally one-to-one. Then the Jacobian conjecture asserts that the function f is both a) globally one-to-one and b) onto. (In other words, that f is a bijection of C2 to itself.) Neither a) nor b) has been proved.

While no doubt well intentioned and technically correct, these statements are more likely to mislead than to clarify the JC for an unsophisticated enquirer.

Here are some reasons:

1. The JC is about a constant JD and a polynomial inverse over a field of characteristic zero and not about algebraically closed fields and local vs. global bijectivity.

2. Powerful mathematics is tacitly assumed without mention. A theorem of Rubin states that a holomorphic map that is locally bijective at a point p has JD that is not zero at p. Then one has to use the fact that a nowhere vanishing polynomial is constant. Neither is true for real numbers (consider y = x^3). Furthermore, a global inverse is not necessarily polynomial (x = y^(1/3)). One needs that, in the complex case, injectivity implies that the map is birational, and hence polynomial [Keller 1939]. Altogether, the particular situation is spectacularly unrepresentative.

3. The insert is subsumed by the later sentence: Even the two variable case has resisted all efforts.

4. Notation is used (z,w) only in the insert.

Recommendation: Find the author and convince him/her to delete it, avoiding any sort of war.

However, I do not know how to do that.

P.S. The JC article probably deserves to have a separate section on the two variable case, which had special techniques and results.

If some generous soul is willing to speak to me briefly about how to use Wikipedia, please email and I will reply with my phone number.

L.Andrew Campbell (talk) 06:03, 13 November 2015 (UTC)


 * This text has been inserted by, and this way of mention him will notify him this discussion automatically. I agree that this text is misplaced in the lead, and I'll remove it. Moreover, this text is unsourced, and, at least partly, WP:Original research. In any case, the first sentence of the text is an editor's opinion and has not its place in Wikipedia. Therefore, reinserting this text in the body of the article would need to edit it dramatically, and expand it by using your explanations.


 * I agree also that a section about the bivariate case would be useful. Could you write it? If you do this, do not worry about a wrong use of Wikipediia: if your edits are useful, but not in Wikipedia standard style, editors that watch this article will probably correct the added content instead of removing it. Some links indicating how to use WP have been added to your talk page by another user.


 * By the way, I suggest also to add in the lead a brief statement of the conjecture. D.Lazard (talk) 10:35, 18 November 2015 (UTC)

Symmetric Case
The current description of the JC for maps with a symmetric Jacobian matrix should be sharpened to the best known results. I propose the following changes, which I hope someone will implement.

There are 3 components to the changes:  modifications to the text in the results section, a new bibliographic reference, and a minor Wikipedia editing issue.

At the bottom of this note is a preliminary version of how the new results section should appear. There are only 3 sentences that differ from their current form. They are the sentences that now contain either the term cubic homogeneous type, or the term cubic linear type, or both terms.

The new reference is as follows:

de Bondt, Michiel; van den Essen, Arno. The Jacobian conjecture for symmetric Druzkowski mappings. Ann. Polon. Math. 86 (2005), no. 1, 43--46. MR2183036 (2006i:14069)

The issue is that the citation of de Bondt and van den Essen at the start of the 3rd modified sentence should now refer to both of their joint papers in 2005, and I don't know the proper style to denote that.

The preliminary  modified results section is as follows:

Results

Wang (1980) proved the Jacobian conjecture for polynomials of degree 2, and Bass, Connell & Wright (1982) showed that the general case follows from the special case where the polynomials are of degree 3, or even more specifically of cubic homogeneous type, meaning F = (X1 + H1, ..., Xn + Hn), where each Hi is either zero or a homogeneous cubic. Druzkowski (1983) showed that one may further assume that the maps are of cubic linear type, meaning the nonzero Hi are cubes of homogeneous linear polynomials. These reductions introduce additional variables and so are not available for fixed N.

Connell & van den Dries (1983) proved that if the Jacobian conjecture is false, then it has a counterexample with integer coefficients and Jacobian determinant 1. In consequence, the Jacobian conjecture is true either for all fields of characteristic 0 or for none. For fixed N, it is true if it holds for at least one algebraically closed field of characteristic 0.

Let k[X] denote the polynomial ring k[X1, ..., Xn] and k[F] denote the k-subalgebra generated by f1, ..., fn. For a given F, the Jacobian conjecture is true if, and only if, k[X] = k[F]. Keller (1939) proved the birational case, that is, where the two fields k(X) and k(F) are equal. The case where k(X) is a Galois extension of k(F) was proved by Campbell (1973) for complex maps and in general by Razar (1979) and, independently, Wright (1981). Adjamagbo (1995) suggested extending the Jacobian conjecture to characteristic p > 0 by adding the hypothesis that p does not divide the degree of the field extension k(X) / k(F). Moh (1983) checked the conjecture for polynomials of degree at most 100 in two variables.

de Bondt & van den Essen (2005) and Druzkowski (2005) independently showed that it is enough to prove the Jacobian Conjecture for complex maps of cubic homogeneous type with a symmetric Jacobian matrix, and further showed that the conjecture holds for maps of cubic linear type  with a symmetric Jacobian matrix over any field of characteristic 0.

The strong real Jacobian conjecture was that a real polynomial map with a nowhere vanishing Jacobian determinant has a smooth global inverse. That is equivalent to asking whether such a map is topologically a proper map, in which case it is a covering map of a simply connected manifold, hence invertible. Sergey Pinchuk (1994) constructed two variable counterexamples of total degree 25 and higher.

P.K. Adjamagbo and A. van den Essen (2007) and Alexei Belov-Kanel and Maxim Kontsevich (2007) showed that the Jacobian conjecture is equivalent to the Dixmier conjecture.

L.Andrew Campbell (talk) 02:37, 13 January 2016 (UTC)

Sentence without referent
One sentence reads:

"A self-contained and purely algebraic proof of the last implication is also given by P. K. Adjamagbo and A. van den Essen (2007) who also proved in the same paper that these two conjectures is equivalent to Poisson conjecture."

In addition to being ungrammatical (the last three words), there is no article referenced called the "Poisson conjecture".

There is no point in contributing to Wikipedia if what you write cannot be understood by other people. I strongly suggest that either someone knowledgeable about this subject and the "Poisson conjecture" explain what it means, or else this sentence be deleted as soon as possible.2601:200:C080:630:294B:3BD8:3731:CC7B (talk) 04:46, 8 January 2021 (UTC)

Inconsistent terminology?
The article introduces the terminology "polynomial mapping", but then goes on to talk about "polynomial (inverse) functions" and "regular functions (all components are polynomials)". I believe all of these mean the same, but I could be wrong. Can we stick to "polynomial mapping" only? MWinter4 (talk) 21:42, 8 February 2024 (UTC)