The bifurcation set of a complex polynomial function of two variables and the Newton polygons of singularities at infinity

Abstract

A. Némethi and A. Zaharia have defined the explicit set for a complex polynomial function $f$ : $C^n\to C$ and conjectured that the bifurcation set of the global fibration of $f$ is given by the union of the set of critical values and the explicit set of $f$. They have proved only the case $n=2$ and $f$ is Newton non-degenerate. In the present paper we will prove this for the case $n=2$, containing the Newton degenerate case, by using toric modifications and give an expression of the bifurcation set of $f$ in the words of Newton polygons.