  We consider non-singular and Jacobian maps whose components are polynomial in
the variable y. We prove that if a map has y-degree one, then it is the
composition of a triangular map and a quasi-triangular map. We also prove that
non-singular y-quadratic maps are injective if one of the leading functional
coefficients does not vanish. Moreover, y-quadratic Jacobian maps are the
composition of a quasi-triangular map and 3 triangular maps. Other results are
given for wider classes of non-singular maps, considering also injectivity on
vertical strips I x R.
