CONTRACTIBLE FIBERS OF POLYNOMIAL FUNCTIONS

In this short note, we investigate the topology of complex polynomials f(x; y) in two variables. The description of the topology of the corresponding level curves Ct:f(x; y) = t is directly related to the vanishing of the leading coeffcients cj(t) of the discriminant of the polynomial f(x; y) - t, regarded as polynomials in t.In particular, we look for condition such that Ct is smooth and con-tractible, since this implies that the corresponding polynomial is a coordinate on C2. A detailed description including the homotopy types of the curves Ct and the associated geometry is so far known only up to the degree 2, and we will try to extend it to the degree 3.