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

TL;DR

This paper characterizes the bifurcation set of real bivariate polynomials using Newton polygons, counts specific singularities at infinity, and provides an upper bound based on toric geometry techniques.

Contribution

It introduces a method to determine bifurcation sets and count singularities at infinity for non-degenerate polynomials using Newton polygons and toric modifications.

Findings

Determined the bifurcation set for non-degenerate polynomials.

Counted 'cleaving' and 'vanishing' singularities at infinity.

Provided an upper bound for the bifurcation set size.

Abstract

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.