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{{Short description|On invertibility of polynomial maps (mathematics)}}
{{More footnotes|date=September 2020}}
{{Infobox mathematical statement
| name = Jacobian conjecture
| image = 
| caption = 
| field = [[Algebraic geometry]]
| conjectured by = [[Ott-Heinrich Keller]]
| conjecture date = 1939
| first proof by = 
| first proof date = 
| open problem = 
| known cases =
| implied by =
| equivalent to = [[Dixmier conjecture]]
| generalizations = 
| consequences =
}}

In [[mathematics]], the '''Jacobian conjecture''' is a famous unsolved problem concerning [[polynomial]]s in several [[Variable (mathematics)|variables]]. It states that if a polynomial function from an ''n''-dimensional space to itself has Jacobian determinant which is a non-zero constant, then the function has a polynomial inverse. It was first conjectured in 1939 by [[Ott-Heinrich Keller]],<ref>{{Citation
| last1=Keller | first1=Ott-Heinrich
| title=Ganze Cremona-Transformationen
| doi=10.1007/BF01695502 | doi-access=free
| year=1939
| journal=Monatshefte für Mathematik und Physik
| issn=0026-9255
| volume=47
| issue=1
| pages=299–306}}</ref> and widely publicized by [[Shreeram Abhyankar]], as an example of a difficult question in [[algebraic geometry]] that can be understood using little beyond a knowledge of [[calculus]].

The Jacobian conjecture is notorious for the large number of attempted proofs that turned out to contain subtle errors. As of 2018, there are no plausible claims to have proved it. Even the two-variable case has resisted all efforts. There are currently no known compelling reasons for believing the conjecture to be true, and according to van den Essen<ref name="vdE1997">{{citation
| last=van den Essen | first= Arno
| chapter= Polynomial automorphisms and the Jacobian conjecture
| title= Algèbre non commutative, groupes quantiques et invariants (Reims, 1995)
| pages= 55–81
| series= Sémin. Congr.
| volume= 2
| publisher= Soc. Math. France
| place= Paris
| year= 1997
| mr=1601194
| chapter-url=https://www.emis.de/journals/SC/1997/2/pdf/smf_sem-cong_2_55-81.pdf}}</ref> there are some suspicions that the conjecture is in fact false for large numbers of variables (indeed, there is equally also no compelling evidence to support these suspicions).  The Jacobian conjecture is number 16 in [[Smale's problems|Stephen Smale's 1998 list of Mathematical Problems for the Next Century]].

==The Jacobian determinant==
Let ''N'' > 1 be a fixed integer and consider  polynomials ''f''<sub>1</sub>, ..., ''f''<sub>''N''</sub> in variables ''X''<sub>1</sub>, ..., ''X''<sub>''N''</sub> with [[coefficient]]s  in a field ''k''. Then we define a [[vector-valued function]] ''F'': ''k<sup>N</sup>'' → ''k''<sup>''N''</sup> by setting:

: ''F''(''X''<sub>1</sub>, ..., ''X''<sub>''N''</sub>) = (''f''<sub>1</sub>(''X''<sub>1</sub>, ...,''X''<sub>''N''</sub>),..., ''f''<sub>''N''</sub>(''X''<sub>1</sub>,...,''X''<sub>''N''</sub>)).

Any map ''F'': ''k<sup>N</sup>'' → ''k''<sup>''N''</sup> arising in this way is called a [[polynomial mapping]].

The [[Jacobian matrix and determinant|Jacobian determinant]] of ''F'', denoted by ''J<sub>F</sub>'', is defined as the [[determinant]] of the ''N'' × ''N'' [[Jacobian matrix]] consisting of the [[partial derivative]]s of ''f<sub>i</sub>'' with respect to ''X<sub>j</sub>'':

:<math>J_F = \left | \begin{matrix} \frac{\partial f_1}{\partial X_1} & \cdots & \frac{\partial f_1}{\partial X_N} \\
\vdots & \ddots & \vdots \\
\frac{\partial f_N}{\partial X_1} & \cdots & \frac{\partial f_N}{\partial X_N} \end{matrix} \right |,</math>

then ''J<sub>F</sub>'' is itself a polynomial function of the ''N'' variables ''X''<sub>1</sub>, ..., ''X<sub>N</sub>''.

==Formulation of the conjecture==
It follows from the multivariable chain rule that if ''F'' has a polynomial inverse function ''G'': ''k<sup>N</sup>'' → ''k<sup>N</sup>'', then ''J<sub>F</sub>'' has a polynomial reciprocal, so is a nonzero constant. The Jacobian conjecture is the following partial converse:

<blockquote>'''Jacobian conjecture:''' Let ''k'' have [[Characteristic (algebra)|characteristic]] 0. If ''J<sub>F</sub>'' is a non-zero constant, then ''F'' has an inverse function ''G'': ''k<sup>N</sup>'' → ''k<sup>N</sup>'' which is [[regular map (algebraic geometry)|regular]], meaning its components are polynomials.</blockquote>

According to van den Essen,<ref name="vdE1997"/> the problem was first conjectured by Keller in 1939 for the limited case of two variables and integer coefficients.

The obvious analogue of the Jacobian conjecture fails if ''k'' has characteristic ''p''&nbsp;>&nbsp;0 even for one variable. The characteristic of a field, if it is not zero, must be prime, so at least 2. The polynomial {{math|''x'' − ''x''<sup>''p''</sup>}} has derivative {{math|1 − ''p x''<sup>''p''−1</sup>}} which is 1 (because ''px'' is 0) but it has no inverse function. However, {{ill|Kossivi Adjamagbo|ht|Pascal Kossivi Adjamagbo}} suggested extending the Jacobian conjecture to characteristic {{nowrap|''p'' > 0}} by adding the hypothesis that ''p'' does not divide the [[Degree of a field extension|degree]] of the [[field extension]] {{nowrap|''k''(''X'') / ''k''(''F'')}}.<ref>{{citation
| mr=1352692
| last=Adjamagbo | first=Kossivi
| chapter=On separable algebras over a U.F.D. and the Jacobian conjecture in any characteristic
| title= Automorphisms of affine spaces (Curaçao, 1994)
| pages= 89–103
| publisher= Kluwer Acad. Publ.
| place= Dordrecht
| year= 1995
| doi=10.1007/978-94-015-8555-2_5 | isbn=978-90-481-4566-9 | doi-access=free}}</ref>

The existence of a polynomial inverse is obvious if ''F'' is simply a set of functions linear in the variables, because then the inverse will also be a set of linear functions. A simple non-linear example is given by

:<math>u=x^2+y+x</math>
:<math>v=x^2+y</math>

so that the Jacobian determinant is

:<math>J_F = \left | \begin{matrix} 1+2x & 1 \\
2x & 1 \end{matrix} \right |
= (1+2x)(1) - (1)2x = 1.
</math>

In this case the inverse exists as the polynomials

:<math>x=u-v</math>
:<math>y=v-(u-v)^2.</math>

But if we modify ''F'' slightly, to

:<math>u=2x^2+y</math>
:<math>v=x^2+y</math>

then the determinant is

:<math>J_F = \left | \begin{matrix} 4x & 1 \\
2x & 1 \end{matrix} \right |
= (4x)(1) - 2x(1) = 2x,
</math>

which is not constant, and the Jacobian conjecture does not apply.
The function still has an inverse:

:<math>x=\sqrt{u-v}</math>
:<math>y=2v-u,</math>

but the expression for ''x'' is not a polynomial.

The condition ''J<sub>F</sub>'' ≠ 0 is related to the [[inverse function theorem]] in [[multivariable calculus]]. In fact for smooth functions (and so in particular for polynomials) a smooth local inverse function to ''F'' exists at every point where ''J<sub>F</sub>'' is non-zero. For example, the map x → ''x''&nbsp;+&nbsp;''x''<sup>3</sup> has a smooth global inverse, but the inverse is not polynomial.

==Results==
Stuart Sui-Sheng Wang proved the Jacobian conjecture for polynomials of [[degree of a polynomial|degree]] 2.<ref>{{Citation
| last1=Wang | first1=Stuart Sui-Sheng
| title=A Jacobian criterion for separability
| journal=Journal of Algebra
| volume=65
| issue=2
| pages=453–494
| date=August 1980
| doi=10.1016/0021-8693(80)90233-1 | doi-access=free }}</ref> Hyman Bass, Edwin Connell, and David Wright showed that the general case follows from the special case where the polynomials are of degree&nbsp;3, or even more specifically, of cubic homogeneous type, meaning  of the form ''F''&nbsp;=&nbsp;(''X''<sub>1</sub>&nbsp;+&nbsp;''H''<sub>1</sub>,&nbsp;...,&nbsp;''X''<sub>''n''</sub>&nbsp;+&nbsp;''H''<sub>''n''</sub>), where each ''H''<sub>''i''</sub> is either zero or a [[Homogeneous polynomial|homogeneous]] cubic.<ref name="BCW1982">{{Citation
| last1=Bass | first1=Hyman
| last2=Connell | first2=Edwin H.
| last3=Wright | first3=David
| title=The Jacobian conjecture: reduction of degree and formal expansion of the inverse
| doi=10.1090/S0273-0979-1982-15032-7
| issn=1088-9485
| mr=663785
| year=1982
| journal=Bulletin of the American Mathematical Society
| series=New Series
| volume=7
| issue=2
| pages=287–330
| doi-access=free}}</ref> Ludwik Drużkowski showed that one may further assume that  the map is of cubic linear type, meaning that the nonzero ''H''<sub>''i''</sub> are cubes of homogeneous linear polynomials.<ref>{{citation
| last=Drużkowski | first=Ludwik M.
| title=An effective approach to Keller's Jacobian conjecture
| journal=[[Mathematische Annalen]]
| volume= 264
| year=1983
| issue=3
| pages=303–313
| mr=0714105
| doi=10.1007/bf01459126 | doi-access=free}}</ref> It seems that Drużkowski's reduction is one most promising way to go forward. These reductions introduce additional variables and so are not available for fixed ''N''.

Edwin Connell and Lou van den Dries proved that if the Jacobian conjecture is false, then it has a counterexample with integer coefficients and Jacobian determinant 1.<ref>{{citation
| last1=Connell | first1=Edwin
| last2=van den Dries | first2=Lou
| title=Injective polynomial maps and the Jacobian conjecture
| journal=Journal of Pure and Applied Algebra
| volume=28
| year=1983
| issue=3
| pages=235–239
| mr=0701351
| doi=10.1016/0022-4049(83)90094-4 | doi-access=free}}</ref> In consequence, the Jacobian conjecture is true either for all fields of characteristic 0 or for none.  For fixed dimension ''N'', it is true if it holds for at least one [[algebraically closed field]] of characteristic 0.

Let ''k''[''X''] denote the [[polynomial ring]] {{nowrap|''k''[''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub>]}} and ''k''[''F''] denote the ''k''-subalgebra generated by ''f''<sub>1</sub>, ..., ''f''<sub>''n''</sub>.  For a given ''F'', the Jacobian conjecture is true if, and only if,  {{nowrap|''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 Andrew Campbell for complex maps<ref>{{citation
| last=Campbell | first=L. Andrew
| title=A condition for a polynomial map to be invertible
| journal=Mathematische Annalen
| volume=205
| issue=3
| year=1973
| pages=243–248
| mr=0324062
| doi=10.1007/bf01349234 | doi-access=free}}</ref> and in general by Michael Razar<ref>{{citation
| last1 = Razar | first1 = Michael
| year = 1979
| title = Polynomial maps with constant Jacobian
| journal = [[Israel Journal of Mathematics]]
| volume = 32
| issue = 2–3
| pages = 97–106
| mr=0531253
| doi=10.1007/bf02764906 | doi-access=free}}</ref> and, independently, by David Wright.<ref>{{citation
| last1 = Wright | first1 = David
| year = 1981
| title = On the Jacobian conjecture
| journal = Illinois Journal of Mathematics
| volume = 25
| issue = 3
| pages = 423–440
| mr=0620428
| doi=10.1215/ijm/1256047158 | doi-access=free}}</ref> Tzuong-Tsieng Moh checked the conjecture for polynomials of degree at most 100 in two variables.<ref>{{citation
| last1=Moh | first1=Tzuong-Tsieng
| title=On the Jacobian conjecture and the configurations of roots
| url=http://resolver.sub.uni-goettingen.de/purl?GDZPPN002200376
| mr=691964
| year=1983
| journal=[[Journal für die reine und angewandte Mathematik]]
| issn=0075-4102
| volume=1983
| issue=340
| pages=140–212
| doi=10.1515/crll.1983.340.140| s2cid=116143599
}}</ref><ref>{{citation
| last1=Moh | first1=Tzuong-Tsieng
| title=On the global Jacobian conjecture for polynomials of degree less than 100
|series=preprint}}</ref>

Michiel de Bondt and Arno van den Essen<ref>{{citation
| last1=de Bondt | first1=Michiel
| last2=van den Essen | first2=Arno
| title=A reduction of the Jacobian conjecture to the symmetric case
| journal=Proceedings of the American Mathematical Society
| volume=133
| year=2005
| issue=8
| pages=2201–2205
| mr=2138860
| doi= 10.1090/S0002-9939-05-07570-2 |doi-access=free | hdl=2066/33302
| hdl-access=free
}}</ref><ref>{{citation
| last1=de Bondt | first1=Michiel
| last2=van den Essen | first2=Arno
| title=The Jacobian conjecture for symmetric Drużkowski mappings
| journal=[[Annales Polonici Mathematici]]
| volume=86
| year=2005
| issue=1
| pages=43–46
| mr=2183036
| doi=10.4064/ap86-1-5 | doi-access=free}}</ref> and Ludwik Drużkowski<ref>{{citation
| last=Drużkowski | first=Ludwik M.
| title=The Jacobian conjecture: symmetric reduction and solution in the symmetric cubic linear case
| journal=Annales Polonici Mathematici
| volume=87
| year=2005
| pages=83–92
| mr=2208537
| doi=10.4064/ap87-0-7 | doi-access=free}}</ref> 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 constructed two variable counterexamples of total degree 35 and higher.<ref>{{citation
| last=Pinchuk | first= Sergey
| title= A counterexample to the strong real Jacobian conjecture
| journal= Mathematische Zeitschrift
| volume= 217
| year=1994
| issue= 1
| pages= 1–4
| mr=1292168
| doi=10.1007/bf02571929 | doi-access=free}}</ref>

It is well known that the [[Dixmier conjecture]] implies the Jacobian conjecture.<ref name="BCW1982"/> Conversely, it is shown by Yoshifumi Tsuchimoto<ref>{{citation
| last=Tsuchimoto | first=Yoshifumi
| year=2005
| title=Endomorphisms of Weyl algebra and <math>p</math>-curvatures
| url=http://projecteuclid.org/euclid.ojm/1153494387
| journal=Osaka Journal of Mathematics
| volume=42
| issue=2
| pages=435–452
| issn=0030-6126}}</ref> and independently by Alexei Belov-Kanel and [[Maxim Kontsevich]]<ref>{{citation
| last1=Belov-Kanel | first1=Alexei
| last2=Kontsevich | first2=Maxim
| title=The Jacobian conjecture is stably equivalent to the Dixmier conjecture
| arxiv=math/0512171
| mr=2337879
| year=2007
| journal=Moscow Mathematical Journal
| volume=7
| issue=2
| pages=209–218
| bibcode=2005math.....12171B
| doi=10.17323/1609-4514-2007-7-2-209-218| s2cid=15150838
}}</ref> that the Jacobian conjecture for ''2N'' variables implies the Dixmier conjecture in ''N'' dimensions. A self-contained and purely algebraic proof of the last implication is also given by Kossivi Adjamagbo and Arno van den Essen<ref>{{citation
| last1= Adjamagbo | first1=Pascal Kossivi
| title= A proof of the equivalence of the Dixmier, Jacobian and Poisson conjectures
| url=http://journals.math.ac.vn/acta/pdf/0702205.pdf
| year=2007
| last2= van den Essen | first2= Arno
| journal= Acta Mathematica Vietnamica
| volume= 32
| pages= 205–214
| mr=2368008}}</ref> who also proved in the same paper that these two conjectures are equivalent to the Poisson conjecture.{{clarification needed|date=December 2024}}

== See also ==

* [[List of unsolved problems in mathematics]]

==References==
{{reflist}}

==External links==
*[http://www.math.purdue.edu/~ttm/jacobian.html Web page of Tzuong-Tsieng Moh on the conjecture]

{{DEFAULTSORT:Jacobian Conjecture}}
[[Category:Polynomials]]
[[Category:Algebraic geometry]]
[[Category:Conjectures]]
[[Category:Unsolved problems in geometry]]