Classification at infinity of polynomials of degree 3 in 3 variables

Abstract

We classify singularities at infinity of polynomials of degree 3 in 3 variables, f(x0, x1, x2) = f1(x0, x1, x2) + f2(x0, x1, x2) + f3(x0, x1, x2), fi homogeneous polynomial of degree i, i = 1, 2, 3. Based on this classification, we calculate the jump in the Milnor number of an isolated singularity at infinity, when we pass from the special fiber to a generic fiber. As an application of the results, we investigate the existence of global fibrations of degree 3 polynomials in C and search for information about the topology of the fibers in each equivalence class.