A note on the Jacobian conjecture
Authors:
Christopher I. Byrnes and Anders Lindquist
Journal:
Proc. Amer. Math. Soc. 136 (2008), 3007-3011
MSC (2000):
Primary 14R15, 55M35; Secondary 47H10
DOI:
https://doi.org/10.1090/S0002-9939-08-09245-9
Published electronically:
April 23, 2008
MathSciNet review:
2407061
Full-text PDF
Abstract | References | Similar Articles | Additional Information
Abstract: In this paper we consider the Jacobian conjecture for a map 
of complex affine spaces of dimension 
. It is well known that if 
is proper, then the conjecture will hold. Using topological arguments, specifically Smith theory, we show that the conjecture holds if and only if 
is proper onto its image.
- 1. S. S. Abyhankar, Expansion techniques in algebraic geometry, Tata Inst. Fundamental Research, Bombay, 1977.
- 2. Hyman Bass, Edwin H. Connell, and David Wright, The Jacobian conjecture: reduction of degree and formal expansion of the inverse, Bull. Amer. Math. Soc. (N.S.) 7 (1982), no. 2, 287–330. MR 663785, https://doi.org/10.1090/S0273-0979-1982-15032-7
- 3. Andrzej Białynicki-Birula and Maxwell Rosenlicht, Injective morphisms of real algebraic varieties, Proc. Amer. Math. Soc. 13 (1962), 200–203. MR 0140516, https://doi.org/10.1090/S0002-9939-1962-0140516-5
- 4. Raoul Bott and Loring W. Tu, Differential forms in algebraic topology, Graduate Texts in Mathematics, vol. 82, Springer-Verlag, New York-Berlin, 1982. MR 658304
- 5. L. Andrew Campbell, A condition for a polynomial map to be invertible, Math. Ann. 205 (1973), 243–248. MR 0324062, https://doi.org/10.1007/BF01349234
- 6. Marvin J. Greenberg, Lectures on algebraic topology, W. A. Benjamin, Inc., New York-Amsterdam, 1967. MR 0215295
- 7. I. N. Herstein, Topics in algebra, Blaisdell Publishing Co. Ginn and Co. New York-Toronto-London, 1964. MR 0171801
- 8. Ott-Heinrich Keller, Ganze Cremona-Transformationen, Monatsh. Math. Phys. 47 (1939), no. 1, 299–306 (German). MR 1550818, https://doi.org/10.1007/BF01695502
- 9. David Mumford, Algebraic geometry. I, Springer-Verlag, Berlin-New York, 1976. Complex projective varieties; Grundlehren der Mathematischen Wissenschaften, No. 221. MR 0453732
- 10. P. A. Smith, Transformations of finite period, Ann. of Math. (2) 39 (1938), no. 1, 127–164. MR 1503393, https://doi.org/10.2307/1968718
Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 14R15, 55M35, 47H10
Retrieve articles in all journals with MSC (2000): 14R15, 55M35, 47H10
Additional Information
Christopher I. Byrnes
Affiliation:
Department of Electrical and Systems Engineering, Washington University in St. Louis, St. Louis, Missouri 63130
Anders Lindquist
Affiliation:
Department of Mathematics, Royal Institute of Technology, 100 44 Stockholm, Sweden
DOI:
https://doi.org/10.1090/S0002-9939-08-09245-9
Keywords:
Jacobian conjecture,
Smith theory.
Received by editor(s):
October 25, 2006
Published electronically:
April 23, 2008
Additional Notes:
This research was supported in part by grants from AFOSR, NSF, the Swedish Research Council, and the Göran Gustafsson Foundation.
Communicated by:
Paul Goerss
Article copyright:
© Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
