相关论文: Unfolding polynomial maps at infinity

We define a new topological invariant of line arrangements in the complex projective plane. This invariant is a root of unity defined under some combinatorial restrictions for arrangements endowed with some special torsion character on the…

Consider a primitive polynomial $f$ in two variables, thought of as a map from the affine plane to the affine line. We study the minimimal compactification of $f$; from our result one deduces in particular that if one of the fibers of $f$…

We study the first homology group of the Milnor fiber of sharp arrangements in the real projective plane. Our work relies on the minimal Salvetti complex of the deconing arrangement and its boundary map. We describe an algorithm which…

We consider the family $\mathrm{MP}_d$ of affine conjugacy classes of polynomial maps of one complex variable with degree $d \geq 2$, and study the map $\Phi_d:\mathrm{MP}_d\to \widetilde{\Lambda}_d \subset \mathbb{C}^d / \mathfrak{S}_d$…

We prove that for any germ of complex analytic set in $\CC^n$ there exists a hypersurface singularity whose Milnor fibration has trivial geometric monodromy and fibre homotopic to the complement of the germ of complex analytic set. As an…

We consider the bifurcation structure of one-parameter families of unimodal maps whose differentiability is only $C^1$. The structure of its bifurcation diagram can be a very wild one in such case. However we prove that in a certain…

Given a regular map f from a smooth complex variety to the affine line, we define a motivic Milnor fiber at infinity and we compute it in the case of a non degenerate Laurent polynomial for its Newton polyhedra at infinity.

We construct a singular variety ${\mathcal{V}}_G$ associated to a polynomial mapping $G : \C^{n} \to \C^{n - 1}$ where $n \geq 2$. We prove that in the case $G : \C^{3} \to \C^{2}$, if $G$ is a local submersion but is not a fibration, then…

We study the monodromy diffeomorphism of Milnor fibrations of isolated complex surface singularities, by computing the family Seiberg--Witten invariant of Seifert-fibered Dehn twists using recent advances in monopole Floer homology. More…

We discuss and prove a number of cohomological results for Milnor fibers, real links, and complex links of local complete intersections with singularities of arbitrary dimension.

We give a concrete formula for the total Milnor number of the weighted-Le-Yomdin-at-infinity polynomial in most of the interesting cases. As an application, we give a description of the monodromy fibration at infinity for such kind of…

The current article studies certain problems related to complex cycles of holomorphic foliations with singularities in the complex plane. We focus on the case when polynomial differential one-form gives rise to a foliation by Riemann…

We prove combinatorial rigidity of infinitely renormalizable unicritical polynomials, P_c :z \mapsto z^d+c, with complex c, under the a priori bounds and a certain "combinatorial condition". This implies the local connectivity of the…

Let $f$ and $g$ be reduced homogeneous polynomials in separate sets of variables. We establish a simple formula that relates the eigenspace decomposition of the monodromy operator on the Milnor fiber cohomology of $fg$ to that of $f$ and…

We study the projection $\pi: M_d \to B_d$ which sends an affine conjugacy class of polynomial $f: \mathbb{C}\to\mathbb{C}$ to the holomorphic conjugacy class of the restriction of $f$ to its basin of infinity. When $B_d$ is equipped with a…

In this paper we prove geometric residue theorems for bundle maps over a compact manifold. The theory developed associates residues to the singularity submanifolds of the map for any invariant polynomial. The theory is then applied to a…

This survey is the continuation of a series of works aimed at applying tools from Singularity Theory to Differential Equations. More precisely, we utilize the powerfull Milnor's Fibration Theory to give geometric-topological classifications…

We show that the natural symplectic structure on the Milnor fiber of an isolated singularity in complex three variables whose link fibers over the circle can be modified into one which is cylindrical at the end. As a consequence we see that…

We consider links of complex isolated hypersurface singularities in $\mathbb{C}^{n+1}$ and study differentiable maps defined by restricting holomorphic functions to the links. We give an explicit example in which such a restriction gives a…

Let $\A$ be an arrangement of affine lines in $\C^2,$ with complement $\M(\A).$ The (co)homo-logy of $\M(\A)$ with twisted coefficients is strictly related to the cohomology of the Milnor fibre associated to the conified arrangement,…