On the bifurcation set of a polynomial function and Newton boundary. II.

On Vanishing Theorems for Local Systems Associated to Laurent Polynomials

“…Let be a sufficiently small tubular neighborhood of in . Then, by [28, Section 3.5] (see also [16]), for any , there exist isomorphisms Moreover, the level set of in is compact in and intersects transversely for any . Hence, for , we have isomorphisms Moreover, for , we have isomorphisms When decreases passing through one of the critical values of , only the dimensions of and may change, and the other cohomology groups remain the same.…”

“…Let be a sufficiently small tubular neighborhood of in , and for , set . Then, by [28, Section 3.5], for any , there exist isomorphisms Moreover, the level set of in is compact in and intersects transversely for any . For , let be the stratified isolated singular points of the function in such that .…”

Monodromies at infinity of non-tame polynomials

“…If U = C n , b = f (0) and in addition to the conditions in Theorems 4.5 and 4.9 (i.e. dimΓ ∞ (f ) = n and f is non-degenerate at infinity) we assume that for any atypical face γ ≺ Γ ∞ (f ) such that dimγ < n − 1 the γ-part f γ : (C * ) n −→ C of f does not have the critical value b, then the meromorphic extension g of f to the compactification X = X Σ satisfies the above-mentioned property in general (see also Némethi-Zaharia [21], Zaharia [33]). In this case the stratified isolated singular points p l+1 , .…”

The bifurcation set of a complex polynomial function of two variables and the Newton polygons of singularities at infinity

“…The results of Suzuki [15], Ha and Le [5] and Ha and Nguyen [6] are known to be pioneering works in these studies, where geometrical and topological characterizations of atypical values at infinity of complex polynomial maps are given. The Newton polygon is one of the main tools in the study of atypical values at infinity, for instance see [11,10,17,8,14]. Concerning real polynomial functions of two variables, Tibȃr and Zaharia gave a characterization of the bifurcation set in [16] in terms of the first betti number, the Euler characteristic and vanishing and splitting phenomena of atypical fibers over the bifurcation set.…”