Multiplicity of complex hypersurface singularities, Rouche' satellites and Zariski's problem

Multiplicity, regularity and blow-spherical equivalence of complex analytic sets

This paper is devoted to study multiplicity and regularity as well as to present some classifications of complex analytic sets. We present an equivalence for complex analytical sets, namely blow-spherical equivalence and we receive several applications with this new approach. For example, we reduce to homogeneous complex algebraic sets a version of Zariski's multiplicity conjecture in the case of blow-spherical homeomorphism, we give some partial answers to the Zariski's multiplicity conjecture, we show that a blow-spherical regular complex analytic set is smooth and we give a complete classif…

Zariski Structures and Algebraic Geometry

The purpose of this paper is to provide a new account of multiplicity for finite morphisms between smooth projective varieties. Traditionally, this has been defined using commutative algebra in terms of the length of integral ring extensions. In model theory, a different approch to multiplicity was developed by Zilber using the techniques of non-standard analysis. Here, we first reformulate Zilber's method in the language of algebraic geometry and secondly show that, in classical projective situations, the two notions essentially coincide. As a consequence, we can recover intersection theory i…

The link of {f(x,y)+z^n=0} and Zariski's Conjecture

We consider suspension hypersurface singularities of type g=f(x,y)+z^n, where f is an irreducible plane curve singularity. For such germs, we prove that the link of g determines completely the Newton pairs of f and the integer n except for two pathological cases, which can be completely described. Even in the pathological cases, the link and the Milnor number of g determine uniquely the Newton pairs of f and n. In particular, for such g, we verify Zariski's conjecture about the multiplicity. The result also supports the following conjecture formulated in the paper. If the link of an isolated h…

Multiplicity of analytic hypersurface singularities under bi-Lipschitz homeomorphisms

We give partial answers to a metric version of Zariski's multiplicity conjecture. In particular, we prove the multiplicity of complex analytic surface (not necessarily isolated) singularities in $\mathbb{C}^3$ is a bi-Lipschitz invariant.

Some classes of homeomorphisms that preserve multiplicity and tangent cones

In this paper we present some applications of A'Campo-Lê's Theorem and we study some relations between Zariski's Questions A and B. It is presented some classes of homeomorphisms that preserve multiplicity and tangent cones of complex analytic sets. Moreover, we present a class of homeomorphisms that has the multiplicity as an invariant when we consider right equivalence and this class contains many known classes of homeomorphisms that preserve tangent cones. In particular, we present some effective approaches to Zariski's Question A. We show a version of these results looking at infinity. Add…

关于阿诺尔德余秩、扎里斯基重数与切米尔诺纤维问题

On the Arnold corank, the Zariski multiplicity, and the tangent Milnor fiber problems

本文研究阿诺尔德余秩等问题，证明余秩模2的拓扑不变性，得到三维阿诺尔德余秩问题的另一证明，还证明相关弱版本扎里斯基重数猜想与欧拉特征等价，给出广义半比雷斯科尔恩-法姆超曲面重数>2为右拓扑等价不变量。

拟齐次曲面的横截切片、Ruas猜想与Zariski重数猜想

Transverse slices, Ruas' conjecture, and Zariski's multiplicity conjecture for quasihomogeneous surfaces

本文研究了余秩1拟齐次映射芽，引入了$μ_{\mathbf{m},\mathbf{k}}$-极小横截切片概念，借此证明了Ruas猜想的部分情形，给出反例回答了Ruas的问题，并证明了此类芽的1-参数族的Zariski重数猜想。

A Non-Standard Bezout Theorem

This paper provides a non-standard analogue of Bezout's theorem. This is acheived by showing that, in all characteristics, the notion of Zariski multiplicity coincides with intersection multiplicity when we consider the full families of projective degree d and degree e curves. The result is particularly interesting in that it holds even when we consider intersections at singular points of curves or when the curves contain non-reduced components. The proof also provides motivation for the fact that tangency is a definable relation for families of curves inside a non-linear 1-dimensional Zariski…

通过局部不变量研究 $\mathbb C^3$ 中的曲面奇点的扎里斯基等奇异性和

Zariski equisingularity of surface singularities in $\mathbb C^3$ by a local invariant

本文通过定义一个新的局部不变量，研究了 $\mathbb C^3$ 中曲面奇点的扎里斯基等奇异性和性质。

Invariants of Newton non-degenerate surface singularities

We recover the Newton diagram (modulo a natural ambiguity) from the link for any surface hypersurface singularity with non-degenerate Newton principal part whose link is a rational homology sphere. As a corollary, we show that the link determines the embedded topological type, the Milnor fibration, and the multiplicity of such a germ. This proves (even a stronger version of) Zariski's Conjecture about the multiplicity for such a singularity.

奇点的Zariski-Nagata定理与一致Izumi-Rees性质

Zariski-Nagata Theorems for Singularities and the Uniform Izumi-Rees Property

本文引入并探讨Noetherian环中的一致Izumi-Rees性质，应用于重数理论及理想符号幂的包含关系，并证明了域上有限型正规域中符号幂包含关系的常数存在性。

Bi-Lipschitz invariance of the multiplicity

The multiplicity of an algebraic curve $C$ in the complex plane at a point $p$ on that curve is defined as the number of points that occur at the intersection of $C$ with a general complex line that passes close to the point $p$. It is shown that $p$ is a singular point of the curve $C$ if and only if this multiplicity is greater than or equal to 2, in this sense, such an integer number can be considered as a measure of how singular can be a point of the curve $C$. In these notes, we address the classical concept of multiplicity of singular points of complex algebraic sets (not necessarily com…

Topologically equisingular deformations of homogeneous hypersurfaces with line singularities are equimultiple

We prove that if $\{f_t\}$ is a family of line singularities with constant Lê numbers and such that $f_0$ is a homogeneous polynomial, then $\{f_t\}$ is equimultiple. This extends to line singularities a well known theorem of A. M. Gabrièlov and A. G. Kušnirenko concerning isolated singularities. As an application, we show that if $\{f_t\}$ is a topologically $\mathscr{V}$-equisingular family of line singularities, with $f_0$ homogeneous, then $\{f_t\}$ is equimultiple. This provides a new partial positive answer to the famous Zariski multiplicity conjecture for a special class of non-isolated…