Bifurcation values of families of real curves

Abstract

Abstract. In more than two variables, detection of the bifurcation set of polynomial mapping R n → R p , n ≥ p, is a still unsolved problem. In this note we provide a solution for n = p + 1 ≥ 3.

On the Lê–Milnor fibration for real analytic maps

“…The equivalence between (i) and (iv) of Theorem 1.1 has a similar nature of a theorem of [16] and [8], which says that if f : C 2 → C is a polynomial function, then a regular value a ∈ C of f is not in the bifurcation set of f if and only if the Euler characteristic of the fibers χ f −1 (t) is constant for t varying in some neighborhood of a. We mention that C. Joita and M. Tibar gave a real counterpart for that theorem in [11].…”

Atypical values and the Milnor set of real polynomials in two variables

“…Our characterization is based in the detection of the vanishing and splitting phenomena at infinity (as defined in [DJT]) making use of the correspondence given in Theorem 5.3. Let us start by recalling some definitions and statements related to atypical values from [Ti3,JT,DT,DJT].…”

Bifurcation set of multi-parameter families of complex curves

“…The phenomenon of ‘vanishing of components’ has been studied in the real setting of polynomials in and more generally in . It turns out that this is related to the ‘vanishing cycles’ and ‘emerging cycles’ introduced for independent reasons by Meigniez .…”

“…Yet another phenomenon which may occur is the ‘splitting at infinity at ’ (in the terminology of ) when approaching the bifurcation value . The simplest example is the complex polynomial .…”