A dominant polynomial map from the complex plane to itself
gives rise to a finite set of curves and isolated points outside its
image. Z. Jelonek provided an upper bound on the number of such
isolated points that is quadratic in, and depends only on, the degrees
of the polynomials involved.
I will introduce in this talk a large family of dominant nonproper
maps above for which this upper bound is linear in the degrees.
Moreover, I will illustrate constructions proving asymptotical
sharpness up to multiplication by a constant.
