Bifurcation set of multi-parameter families of complex curves

Abstract

The problem of detecting the bifurcation set of polynomial mappings C m → C k , m 2, m k 1, has been solved in the case m = 2, k = 1 only. Its solution, which goes back to the 1970s, involves the non-constancy of the Euler characteristic of fibers. We provide here a complete answer to the general case m = k + 1 3 in terms of the Betti numbers of fibers and of a vanishing phenomenon discovered in the late 1990s in the real setting.

On topological approaches to the Jacobian conjecture in ℂn

“…It is well known that analytic and geometric conditions can be used to estimate B(F k ); see, for instance, [6,15,17,19,25]. Thus, we may use these conditions to ensure the topological hypothesis related to B(F k ) in Theorems 1.6, 1.7 and 2.3.…”