Counting isolated points outside the image of a polynomial map

Abstract

We consider a generic family of polynomial maps $f:=(f_1,f_2):{\mathbb{C}}^2\to {\mathbb{C}}^2$ 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 $6d(f)$. 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 $f$.

24 pages, 5 figures. Final version, accepted for publication in 'Advances in Geometry'. Comments are very welcome!