Counting isolated points outside the image of a polynomial map

Abstract

We consider a generic family of polynomial maps $f:=(f_1,f_2):\mathbb{C}^2\rightarrow\mathbb{C}^2$ with given supports of polynomials, and degree $ d(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 $\mathbb{C}^2\setminus f(\mathbb{C}^2)$ has a size of at most $6 d(f)$. This improves an existing upper bound $(d(f) - 1)^2$ proven by Jelonek. Moreover, for each $n\in\mathbb{N}$, we construct a dominant map $f$ above, with $d(f) = 2n+2$, and having $2n$ isolated points in $\mathbb{C}^2\setminus f(\mathbb{C}^2)$. 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 $f$.