Contractible fibers of polynomial functions

Abstract

In this short note, we investigate the topology of complex polynomials f(x,y)f(x,y) in two variables. The description of the topology of the corresponding level curves Ct:f(x,y)=tCt:f(x,y)=t is directly related to the vanishing of the leading coefficients cj (t) of the discriminant of the polynomial f(x,y)−tf(x,y)−t, regarded as polynomials in tt.