Related papers: Join theorem for real analytic singularities

We consider a real analytic map $F=(f_1,...,f_k) : (\mathbb{R}^n,0) \rightarrow (\mathbb{R}^k,0)$, $2 \le k \le n-1$, that satisfies Milnor's conditions (a) and (b) introduced by D. Massey. This implies that every real analytic…

In this article we study the topology of a family of real analytic germs $F \colon (\mathbb{C}^3,0) \to (\mathbb{C},0)$ with isolated critical point at 0, given by $F(x,y,z)=f(x,y)\bar{g(x,y)}+z^r$, where $f$ and $g$ are holomorphic, $r \in…

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…

For analytic map germs $f: (\mathbb{R}^n, 0)\to (\mathbb{R}, 0)$ having an isolated critical value in the origin with $\dim V(f)>0$ and satisfying the transversality property of D.B. Massey we show that for $c>0$ a large enough constant,…

Let f and g be holomorphic function-germs vanishing at the origin of a complex analytic germ of dimension three. Suppose that they have no common irreducible component and that the real analytic map-germ given by the multiplication of f by…

We study the topology of the boundaries of the Milnor fibers of real analytics map-germs $f: (\mathbb{R}^M,0) \to (\mathbb{R}^K,0)$ and $f_{I}:=\Pi_{I}\circ f : (\mathbb{R}^M,0) \to (\mathbb{R}^I,0)$ that admit Milnor's tube fibrations,…

Budur, Fernandes de Bobadilla, Le and Nguyen (2022) conjectured that if two germs of holomorphic functions are topologically equivalent, then the Milnor fibres of their initial forms are homotopy equivalent. In this note, we give…

We prove that to each real singularity $f: (\mathbb{R}^{n+1}, 0) \to (\mathbb{R}, 0)$ one can associate two systems of differential equations $\mathfrak{g}^{k\pm}_f$ which are pushforwards in the category of $\mathcal{D}$-modules over…

In this paper, we study the topology of real analytic map-germs with isolated critical value $f: (\mathbb{R}^m,0) \to (\mathbb{R}^n,0)$, with $1 <n <m$. We compare the topology of $f$ with the topology of the compositions $\pi_i^* \circ f$,…

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…

We constructed the analytic Milnor fiber is a non-archimedean model of the classical topological Milnor fibration. In the present paper, we describe the homotopy type of the analytic Milnor fiber in terms of a strictly semi-stable model,…

When f : R power n to R power p, is a surjective real analytic map with isolated critical value, we prove that the (m)-regularity condition (in a sense we define) ensures that f ||f|| is a fibration on small spheres, f induces a fibration…

In this paper, we introduce a new regularity condition that characterizes the tameness of a composite singularity $H=G\circ F$ in a sharp way. Our approach provides a natural tool that links the topology of the Milnor tube fibrations…

In this article, we study the topology of real analytic germs $F \colon (\C^3,0) \to (\C,0)$ given by $F(x,y,z)=\overline{xy}(x^p+y^q)+z^r$ with $p,q,r \in \N$, $p,q,r \geq 2$ and $(p,q)=1$. Such a germ gives rise to a Milnor fibration…

We show how formal and rigid geometry can be used in the theory of complex singularities, and in particular in the study of the Milnor fibration and the motivic zeta function. We introduce the so-called analytic Milnor fiber associated to…

We are going to use the Euler's vector fields in order to show that for real quasi-homogeneous singularities with isolated critical value, the Milnor's fibration in a "thin" hollowed tube involving the zero level and the fibration in the…

In this article we prove two results concerning the motivic Milnor fibres $S^{\epsilon}(f)$ associated to a map germ $f: (\mathbb{R}^n,0)\to(\mathbb{R},0)$, defined by G. Comte and G. Fichou. Firstly, we prove that if…

In this article we apply the results in the article "On Isolated Real Singularities I" to the study of real $ADE$-singularities. We show that said results enables us to find the homology groups of the Milnor fibres of real…

Let $X$ be an analytic subset of an open neighbourhood $U$ of the origin $\underline{0}$ in $\mathbb{C}^n$. Let $f\colon (X,\underline{0}) \to (\mathbb{C},0)$ be holomorphic and set $V =f^{-1}(0)$. Let $\B_\epsilon$ be a ball in $U$ of…

Let (X_R, 0) be a germ of real analytic subset in (R^N, 0) of pure dimension n+1 with an isolated singularity at 0. Let (f_R,0) : (X_R, 0) --> (R,0) a real analytic germ with an isolated singularity at 0, such that its complexification f_C…