  Jedrzejewicz showed that a polynomial map over a field of characteristic zero
is invertible, if and only if the corresponding endomorphism maps irreducible
polynomials to irreducible polynomials. Furthermore, he showed that a
polynomial map over a field of characteristic zero is a Keller map, if and only
if the corresponding endomorphism maps irreducible polynomials to square-free
polynomials. We show that the latter endomorphism maps other square-free
polynomials to square-free polynomials as well.
  In connection with the above classification of invertible polynomial maps and
the Jacobian Conjecture, we study irreducible properties of several types of
Keller maps, to each of which the Jacobian Conjecture can be reduced. Herewith,
we generalize the result of Bakalarski, that the components of cubic
homogeneous Keller maps with a symmetric Jacobian matrix (over C and hence any
field of characteristic zero) are irreducible.
  Furthermore, we show that the Jacobian Conjecture can even be reduced to any
of these types with the extra condition that each affinely linear combination
of the components of the polynomial map is irreducible. This is somewhat
similar to reducing the planar Jacobian Conjecture to the so-called (planar)
weak Jacobian Conjecture by Kaliman.
