Test polynomials, retracts, and the Jacobian conjecture

Abstract

Let K[x,y] be the algebra of two-variable polynomials over a field K. A polynomial p=p(x, y) is called a test polynomial (for automorphisms) if, whenever \phi(p)=p for a mapping \phi of K[x,y], this \phi must be an automorphism. Here we show that p \in C[x,y] is a test polynomial if and only if p does not belong to any proper retract of C[x,y]. This has the following corollary that may have application to the Jacobian conjecture: if a mapping \phi of C[x,y] with invertible Jacobian matrix is ``invertible on one particular polynomial", then it is an automorphism. More formally: if there is a non-constant polynomial p and an injective mapping \psi of C[x,y] such that \psi(\phi(p)) =p, then \phi is an automorphism.