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Bernstein-Sato Polynomials and Spectral Numbers

Andréa G. Guimarães, Abramo Hefez (2007)

Annales de l’institut Fourier

In this paper we will describe a set of roots of the Bernstein-Sato polynomial associated to a germ of complex analytic function in several variables, with an isolated critical point at the origin, that may be obtained by only knowing the spectral numbers of the germ. This will also give us a set of common roots of the Bernstein-Sato polynomials associated to the members of a μ -constant family of germs of functions. An example will show that this set may sometimes consist of all common roots.

Invertible polynomial mappings via Newton non-degeneracy

Ying Chen, Luis Renato G. Dias, Kiyoshi Takeuchi, Mihai Tibăr (2014)

Annales de l’institut Fourier

We prove a sufficient condition for the Jacobian problem in the setting of real, complex and mixed polynomial mappings. This follows from the study of the bifurcation locus of a mapping subject to a new Newton non-degeneracy condition.

On the Hyperbolicity Domain of the Polynomial x^n + a1x^(n-1) + 1/4+ an

Kostov, Vladimir (1999)

Serdica Mathematical Journal

∗ Partially supported by INTAS grant 97-1644We consider the polynomial Pn = x^n + a1 x^(n−1) + · · · + an , ai ∈ R. We represent by figures the projections on Oa1 . . . ak , k ≤ 6, of its hyperbolicity domain Π = {a ∈ Rn | all roots of Pn are real}. The set Π and its projections Πk in the spaces Oa1 . . . ak , k ≤ n, have the structure of stratified manifolds, the strata being defined by the multiplicity vectors. It is known that for k > 2 every non-empty fibre of the projection Π^k → Π^(k−1) is...

Quadratic mappings and configuration spaces

Gia Giorgadze (2003)

Banach Center Publications

We discuss some approaches to the topological study of real quadratic mappings. Two effective methods of computing the Euler characteristics of fibers are presented which enable one to obtain comprehensive results for quadratic mappings with two-dimensional fibers. As an illustration we obtain a complete topological classification of configuration spaces of planar pentagons.

Singular holomorphic functions for which all fibre-integrals are smooth

D. Barlet, H. Maire (2000)

Annales Polonici Mathematici

For a germ (X,0) of normal complex space of dimension n + 1 with an isolated singularity at 0 and a germ f: (X,0) → (ℂ,0) of holomorphic function with df(x) ≤ 0 for x ≤ 0, the fibre-integrals     s ↦ ∫ f = s ϱ ω ' ⋀ ω ' ' ¯ , ϱ ∈ C c ∞ ( X ) , ω ' , ω ' ' ∈ Ω X n , are C ∞ on ℂ* and have an asymptotic expansion at 0. Even when f is singular, it may happen that all these fibre-integrals are C ∞ . We study such maps and build a family of examples where also fibre-integrals for ω ' , ω ' ' ∈ ⍹ X , the Grothendieck sheaf, are C ∞ .

Singularities, defects and chaos in organized fluids

Roland Ribotta, Ahmed Belaidi, Alain Joets (2003)

Banach Center Publications

The singularities occurring in any sort of ordering are known in physics as defects. In an organized fluid defects may occur both at microscopic (molecular) and at macroscopic scales when hydrodynamic ordered structures are developed. Such a fluid system serves as a model for the study of the evolution towards a strong disorder (chaos) and it is found that the singularities play an important role in the nature of the chaos. Moreover both types of defects become coupled at the onset of turbulence....

Singularities of convex hulls as fronts of Legendre varieties

Ilia Bogaevski (1999)

Banach Center Publications

We describe singularities of the convex hull of a generic compact smooth hypersurface in four-dimensional affine space up to diffeomorphism. It turns out that the boundary of the convex hull is the front of a Legendre variety. Its singularities are classified up to contact diffeomorphism.

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