Related papers: Geometry of non-degenerate locally tame non-isolat…
We investigate the equisingularity question for $1$-parameter deformation families of mixed polynomial functions $f_t(\mathbf{z},\bar{\mathbf{z}})$ from the Newton polygon point of view. We show that if the members $f_t$ of the family…
Convenient mixed functions with strongly non-degenerate Newton boundaries have Milnor fibrations, as the isolatedness of the singularity follows from the convenience. In this paper, we consider the Milnor fibration for non-convenient mixed…
In an unpublished lecture note, J. Brian\c{c}on observed that if $\{f_t\}$ is a family of isolated complex hypersurface singularities such that the Newton boundary of $f_t$ is independent of $t$ and $f_t$ is non-degenerate, then the…
Using Newton polyhedra and non-degeneracy of matrices we present conditions which guarantee the Whitney equisingularity of families of isolated determinantal singularities.
We study the topological triviality and the Whitney equisingularity of a family of isolated determinantal singularities. On one hand, we give a L\^e-Ramanujam type theorem for this kind of singularities by using the vanishing Euler…
The main purpose of this article is to lay the foundations for a classification of isolated hypersurface singularities in positive characteristic. Although our article is in the spirit of Arnol'd who classified real an complex hypersurfaces…
We prove that for two germs of analytic mappings $f,g\colon (\mathbb{C}^n,0) \rightarrow (\mathbb{C}^p,0)$ with the same Newton polyhedra which are (Khovanskii) non-degenerate and their zero sets are complete intersections with isolated…
In this paper, we establish conditions for a family $\{f_t\}$ of functions, with not necessarily isolated singularities, defined on a toric variety so that the associated family of hypersurfaces $\{f_t^{-1}(0)\}$ is Whitney equisingular. We…
We study the topology of a line singularity, which is a complex hypersurface with non-isolated singularity given by a complex line. We describe the degeneration of its Milnor fibre to the singular hypersurface by means of a pair of…
We introduce two generalizations of Newton-non-degenerate (Nnd) singularities of hypersurfaces. Roughly speaking, an isolated hypersurface singularity is called topologically Newton-non-degenerate (tNnd) if the local embedded topological…
In the study of equisingularity of isolated singularities we have the classical theorem of Briancon, Speder and Teissier which states that a family of isolated hypersurface singularities is Whitney equisingular if and only if the…
This paper has been withdrawn. Consider an isolated complex hypersurface singularity, f(x_1,..,x_n)=0. For Newton-non-degenerate singularities the local topology is completely determined by an associated polyhedral object, the Newton…
Given a nonconstant polynomial map over the reals having an isolated critical point in the origin and with zero locus of positive dimension, we establish a formula for the singular homology groups of a Milnor fibre relative to its boundary.
Let f be a hypersurface surface local singularity whose zero set has 1-dimensional singular locus. We develop an explicit procedure that provides the boundary of the Milnor fibre of f as an oriented plumbed 3-manifold. The method provides…
In this work we study equisingularity in a one-parameter flat family of generically reduced curves. We consider some equisingular criteria as topological triviality, Whitney equisingularity and strong simultaneous resolution. In this…
We give in this work an explicit combinatorial algorithm for the description of the Milnor fiber of a Newton non degenerate surface singularity as a graph manifold. This is based on a previous work by the author describing a general method…
We show that every $\mu$-constant family of isolated hypersurface singularities satisfying a nondegeneracy condition in the sense of Kouchnirenko, is topologically trivial, also is equimultiple.
We present new results on equisingularity and equinormalizability of families with isolated non-normal singularities (INNS) of arbitrary dimension. We define a $\delta$-invariant and a $\mu$-invariant for an INNS and prove necessary and…
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 prove fibration theorems \`a la Milnor for differentiable real maps with non isolated critical values. We study the situation for maps with linear discriminant, and prove that the concept of d-regularity is the key point for the…