ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

4. Algebraic Geometry

1991 MS Classification: 13B25, 13P10, 14E09. Yu, Jie-Tai, The University of Hong Kong, Hong Kong, China Polynomial Retracts and the Jacobian Conjecture 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) φ : K[x, y] → K[x, y] such that φ(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. 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. The talk is based on a recent joint paper accepted for publication in Transactions of the American Mathematical Society with Vladimir Shpilrain.