The bifurcation set of a real polynomial function of two variables and   Newton polygons of singularities at infinity
  In this paper, we determine the bifurcation set of a real polynomial function of two variables for non-degenerate case in the sense of Newton polygons by using a toric compactification. We also count the number of singular phenomena at infinity, called "cleaving" and "vanishing" in the same setting. Finally, we give an upper bound of the number of elements in the bifurcation set in terms of its Newton polygon. To obtain the upper bound, we apply toric modifications to the singularities at infinity successively. 
Masaharu Ishikawa, Tat Thang Nguyen, Tien Son Pham
math.GT math.AG