Editing Jacobian conjecture (section)

{{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]].