  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.
