Jacobian Conjecture -- from Wolfram MathWorld

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Jacobian Conjecture

The Jacobian conjecture in the plane states that given a ring map of (the polynomial ring in two variables over the complex numbers ) to itself that fixes and sends , to , respectively, is an automorphism iff the Jacobian is a nonzero element of . The condition can easily shown to be necessary, but proving sufficiency has been an open problem since Keller (1939).

The Jacobian conjecture is one of Smale's problems.

There have been at least five published incorrect proofs and many incorrect attempts over the years. In November 2004, Hochster (2004) sent an email announcing a new proof by Carolyn Dean. However, this proof unfortunately contained an error as well.

REFERENCES:

Abhyankar, S. S. Lectures on Expansion Techniques in Algebraic Geometry. Bombay, India: Tata Institute of Fundamental Research, 1977.

Bass, H. "Conjecture jacobienne et opérateurs différentiels." Mém. Soc. Math. France, No. 38, 39-50, 1989.

Becker, T. and Weispfenning, V. Gröbner Bases: A Computational Approach to Commutative Algebra. New York: Springer-Verlag, p. 330, 1993.

Drużkowski, L. M. "The Jacobian Conjecture." IMPAN Preprint 492. Kraków, Poland: Math. Inst. Jagiellonian University, 1991.

Formanek, E. "Observations About the Jacobian Conjecture." Houston J. Math. 20, 369-380, 1994.

Hochster, M. "Lectures on Jacobian Conjecture." sci.math.research post forwarded by I. Algol. Nov. 11, 2004.

Bass, H.; Connell, E. H.; and Wright, D. "The Jacobian Conjecture: Reduction of Degree and Formal Expansion of the Inverse." Bull. Amer. Math. Soc. 7, 287-330, 1982.

Keller, O.-H. "Ganze Cremona Transformationen." Monatsh. für Math. u. Phys. 47, 299-306, 1939.

Meisters, G. H. "Jacobian Problems in Differential Equations and Algebraic Geometry." Rocky Mountain J. Math. 12, 679-705, 1982.

Meisters, G. H. "Wanted: A Bad Matrix." Amer. Math. Monthly 102, 546-550, 1995.

Smale, S. "Mathematical Problems for the Next Century." Math. Intelligencer 20, No. 2, 7-15, 1998.

Smale, S. "Mathematical Problems for the Next Century." In Mathematics: Frontiers and Perspectives 2000 (Ed. V. Arnold, M. Atiyah, P. Lax, and B. Mazur). Providence, RI: Amer. Math. Soc., 2000.

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Weisstein, Eric W. "Jacobian Conjecture." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/JacobianConjecture.html