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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
Published electronically:
April 23, 2008
MathSciNet review:
2407061
Full-text PDF Free Access
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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
(83k:14028), http://dx.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
(25 #3936), http://dx.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, 1982. MR 658304
(83i:57016)
- 5.
L.
Andrew Campbell, A condition for a polynomial map to be
invertible, Math. Ann. 205 (1973), 243–248. MR 0324062
(48 #2414)
- 6.
Marvin
J. Greenberg, Lectures on algebraic topology, W. A. Benjamin,
Inc., New York-Amsterdam, 1967. MR 0215295
(35 #6137)
- 7.
I.
N. Herstein, Topics in algebra, Blaisdell Publishing Co. Ginn
and Co. New York-Toronto-London, 1964. MR 0171801
(30 #2028)
- 8.
Ott-Heinrich
Keller, Ganze Cremona-Transformationen, Monatsh. Math. Phys.
47 (1939), no. 1, 299–306 (German). MR
1550818, http://dx.doi.org/10.1007/BF01695502
- 9.
David
Mumford, Algebraic geometry. I, Springer-Verlag, Berlin, 1976.
Complex projective varieties; Grundlehren der Mathematischen
Wissenschaften, No. 221. MR 0453732
(56 #11992)
- 10.
P.
A. Smith, Transformations of finite period, Ann. of Math. (2)
39 (1938), no. 1, 127–164. MR
1503393, http://dx.doi.org/10.2307/1968718
- 1.
- S. S. Abyhankar, Expansion techniques in algebraic geometry, Tata Inst. Fundamental Research, Bombay, 1977.
- 2.
- H. Bass, E. H. Connell and D. Wright, The Jacobian conjecture: Reduction of degree and formal expansion of the inverse, Bull. Amer. Math. Soc. (N.S.) 7 (1982), 287-330. MR 663785 (83k:14028)
- 3.
- A. Bialynicki-Birula and M. Rosenlicht, Injective morphisms of real algebraic varieties, Proc. Amer. Math. Soc. 13 (1962), 200-203. MR 0140516 (25:3936)
- 4.
- R. Bott and L. W. Tu, Differential Forms in Algebraic Topology, Springer-Verlag, New York, Heidelberg, Berlin, 1982. MR 658304 (83i:57016)
- 5.
- L. A. Campbell, A condition for a polynomial map to be invertible, Math. Ann. 205 (1973), 243-248. MR 0324062 (48:2414)
- 6.
- M. Greenberg, Lectures on Algebraic Topology, W. A. Benjamin, Inc., New York, 1967. MR 0215295 (35:6137)
- 7.
- I. N. Herstein, Topics in Algebra, Blaisdell Publishing Co., Ginn and Co., New York, Toronto, London, 1964. MR 0171801 (30:2028)
- 8.
- O. H. Keller, Ganze Cremona-Transformationen, Monats. Math. Physik 47 (1939), 299-306. MR 1550818
- 9.
- D. Mumford, Algebraic Geometry I: Complex Projective Varieties, Springer-Verlag, Berlin, Heidelberg, New York, 1976. MR 0453732 (56:11992)
- 10.
- P. A. Smith, Transformations of finite period, Annals of Mathematics (2) 39 (1938), 127-164. MR 1503393
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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:
http://dx.doi.org/10.1090/S0002-9939-08-09245-9
PII:
S 0002-9939(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.
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