Counting isolated points outside the image of a polynomial map

We consider a generic family of polynomial maps f := ( f 1 , f 2 ) : ℂ 2 → ℂ 2 with given supports of polynomials, and degree deg f := max(deg f 1 , deg f 2 ). We show that the (non-) properness of maps f in this family depends uniquely on the pair of supports, and that the set of isolated points in ℂ 2 ∖ f (ℂ 2 ) has a size of at most 6 deg f . This improves an existing upper bound (deg f – 1) 2 proven by Jelonek. Moreover, for each n ∈ ℕ, we construct a dominant map f as above, with deg f = 2 n + 2, and having 2 n isolated points in ℂ 2 ∖ f (ℂ 2 ). Our proofs are constructive and can be adapted to a method for computing isolated missing points of f . As a byproduct, we describe those points in terms of singularities of the bifurcation set of f .