Cyclicity and Maximal Multiplicity for Zeros of Families of Analytic Functions

A Rolle type theorem for cyclicity of zeros of families of analytic functions

Let $\{f_{λ; j}\}_{λ\in V; 1\le j\le k}$ be families of holomorphic functions in the open unit disk $\Di\subset\Co$ depending holomorphically on a parameter $λ\in V\subset \Co^n$. We establish a Rolle type theorem for the generalized multiplicity (called {\em cyclicity}) of zero of the family of univariate holomorphic functions ${\sum_{j=1}^k f_{λ;j}}_{λ\in V}$ at $0\in\Di$. As a corollary, we estimate the cyclicity of the family of generalized exponential polynomials, that is, the family of entire functions of the form $\sum_{k=1}^m P_k(z)e^{Q_k(z)}$, $z\in\Co$, where $P_k$ and $Q_k$ are holo…

On the multiplicities of families of complex hypersurface-germs

We show that the possible drop in multiplicity in an analytic family $F(z,t)$ of complex analytic hypersurface singularities with constant Milnor number is controlled by the powers of $t$. We prove equimultiplicity of $μ$-constant families of the form $f + tg + t^2h$ if the singular set of the tangent cone of $\{f = 0}$ is not contained in the tangent cone of $\{h = 0}$.

Whitney equisingularity in families of generically reduced curves

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 context, we prove that Whitney equisingularity is equivalent to strong simultaneous resolution and it is also equivalent to the constancy of the Milnor number and the multiplicity of the fibers. These results are extensions to the case of flat deformations of generically reduced curves, of known results on reduced curves. When the family $(X,0)$ is topologically t…

曲线的完全交差异 II：曲线族

The Complete Intersection Discrepancy of a Curve II: Families of Curves

本文研究了曲线族的等奇异性和完全交差异，证明了魏特奇等奇异性和强同时解析性的等价性，并展示了米尔诺数的上半连续性及其在拓扑等奇异中的应用。