Counting isolated points outside the image of a polynomial map

TL;DR

This paper establishes a new upper bound on the number of isolated points outside the image of certain polynomial maps from ^2 to ^2, improving previous bounds and providing constructive methods for their computation.

Contribution

It introduces a sharper upper bound of 6 times the degree for isolated points outside the polynomial map image, refining Jelonek's earlier quadratic bound, and offers constructive proofs and methods.

Findings

Upper bound of 6d(f) for isolated points, improving previous (d(f)-1)^2 bound.

Constructed examples with 2n isolated points for degree 2n+2 maps.

Descriptions of isolated points via singularities of bifurcation sets.

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 $f$.