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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Polynomial Retracts
and the Jacobian Conjecture


Authors: Vladimir Shpilrain and Jie-Tai Yu
Journal: Trans. Amer. Math. Soc. 352 (2000), 477-484
MSC (1991): Primary 13B25, 13P10; Secondary 14E09, 16S10
Published electronically: September 21, 1999
MathSciNet review: 1487631
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Abstract: Let $ K[x, y]$ be the polynomial algebra in two variables over a field $K$ of characteristic $0$. A subalgebra $R$ of $K[x, y]$ is called a retract if there is an idempotent homomorphism (a retraction, or projection) $\varphi: K[x, y] \to K[x, y]$ such that $\varphi(K[x, y]) = R.$ The presence of other, equivalent, definitions of retracts provides several different methods of studying and applying them, and brings together ideas from combinatorial algebra, homological algebra, and algebraic geometry. In this paper, we characterize all the retracts of $ K[x, y]$ up to an automorphism, and give several applications of this characterization, in particular, to the well-known Jacobian conjecture.


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Additional Information

Vladimir Shpilrain
Affiliation: Department of Mathematics, The City College of New York, New York, New York 10031
Email: shpil@groups.sci.ccny.cuny.edu

Jie-Tai Yu
Affiliation: Department of Mathematics, University of Hong Kong, Pokfulam Road, Hong Kong
Email: yujt@hkusua.hku.hk

DOI: http://dx.doi.org/10.1090/S0002-9947-99-02251-5
PII: S 0002-9947(99)02251-5
Received by editor(s): March 11, 1997
Received by editor(s) in revised form: August 20, 1997
Published electronically: September 21, 1999
Additional Notes: The second author’s research was partially supported by RGC Fundable Grant 344/024/0002
Article copyright: © Copyright 1999 American Mathematical Society