Bifurcation values of polynomial functions and perverse sheaves

Abstract

We characterize bifurcation values of polynomial functions by using the theory of perverse sheaves and their vanishing cycles. In particular, by introducing a method to compute the jumps of the Euler characteristics with compact support of their fibers, we confirm the conjecture of Némethi-Zaharia in many cases.Résumé. -Nous caractérisons les valeurs de bifurcation de fonctions polynomiales en utilisant la théorie des faisceaux pervers et leurs cycles évanescents. En particulier, en introduisant une méthode pour calculer les sauts de caractéristiques d'Euler à support compact de leurs fibres, nous confirmons la conjecture de Némethi-Zaharia dans de nombreux cas.

Asymptotic critical value set and Newton polyhedron

“…and f W γ (u) = u 3 3 − 3u 3 has singular points u * 3 = ±1 and critical values ∓2 respectively. After [13] we see that in this case the bifurcation set B(f…”

“…Even in this situation we can construct a curve X(t) satisfying (I), (II) of Definition 3.1 approaching to the value f γ (Sing f γ ∩ (C * ) dim γ ) for γ : nonrelatively simple bad face. This gives an example to Corollary 3.1 that is not covered by [13]. It seems that, in comparison with the bifurcation value set, the asymptotic critical value set requires far less conditions to be imposed on the face.…”

“…The main result Theorem 1.6. of [13] relies heavily on the notion of relatively simple faces. It shows that the set γ f γ (Sing f γ ∩ (C * ) dim γ ) where γ runs relatively simple bad faces is contained in the bifurcation value set of a polynomial mapping f under the condition of non-degeneracy and isolated singularities at infinity.…”

“…In [13] it is shown that the left hand side set of the above inclusion relation is contained in the bifurcation set B(f ) for γ relatively simple bad face (see Definition 4.1) and f W γ with isolated singularities on (C * ) dim γ . As it is known B(f ) ⊂ K ∞ (f ) from [8], our Corollary 3.1 represents a new result only for γ non-relatively simple bad face if the condition of the isolated singularities at infinity is assumed.…”

“…In Section 4 we examine an example of a polynomial in 5 variables with non-relatively simple bad face. Even in this situation we can construct a curve approaching to an asymptotic critical value of f. This gives an example to Corollary 3.1 that is not covered by [13]. As [13] imposes the condition of isolated singularity at infinity, it does not concern our Example 5.1 treating the non-isolated singularities.…”