Polynomial Retracts and the Jacobian Conjecture

Shpilrain, Vladimir; Yu, Jie-Tai

doi:10.1090/S0002-9947-99-02251-5

Polynomial Retracts and the Jacobian Conjecture
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by Vladimir Shpilrain and Jie-Tai Yu

Trans. Amer. Math. Soc. 352 (2000), 477-484

DOI: https://doi.org/10.1090/S0002-9947-99-02251-5

Published electronically: September 21, 1999

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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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