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.\bm{C}ⁿ→ \bm{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.