Polynomial retracts and the Jacobian conjecture

Let K[x, y] be the polynomial algebra in two variables over a field K of ch aracteristic 0. A subalgebra R of K[x, y] is called a retract if there is a n idempotent homomorphism (a retraction, or projection) phi : K[x, y] --> K [x, y] such that phi(K[x, y]) = R. The presence of other, equivalent, defin itions of retracts provides several different methods of studying and apply ing them, and brings together ideas from combinatorial algebra, homological algebra, and algebraic geometry. In this paper, we characterize all the re tracts of K[x, y] up to an automorphism, and give several applications of t his characterization, in particular, to the well-known Jacobian conjecture.