## Polynomial Retracts and the Jacobian Conjecture

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- by Vladimir Shpilrain and Jie-Tai Yu PDF
- Trans. Amer. Math. Soc.
**352**(2000), 477-484 Request permission

## 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
- 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
- © Copyright 1999 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**352**(2000), 477-484 - MSC (1991): Primary 13B25, 13P10; Secondary 14E09, 16S10
- DOI: https://doi.org/10.1090/S0002-9947-99-02251-5
- MathSciNet review: 1487631