函数芽的有限R_r~*(S;n)-决定性

光滑映射芽的有限决定性是奇点理论中一个重要专题 .对函数芽的有限决定性问题 ,主要是在右等价群及其一些子群作用下来讨论的 .本文在 [1]和 [4 ]的基础上讨论函数芽在右等价群的正规子群 R*n (S;n)作用下的有限决定性 ,并组出函数芽有限 R*r (S;n) -决定的一个充分必要条件 .     

函数芽的有限R*r(S; n)-决定性

光滑映射芽的有限决定性是奇点理论中一个重要专题．对函数芽的有限决定性问题，主要是在右等价群及其一些子群作用下来讨论的．本文在[1]和[4]的基础上讨论函数芽在右等价群的正规子群Rr^*(S；n)作用下的有限决定性，并组出函数芽有限Rr^*(S；n)-决定的一个充分必要条件．     

光滑映射芽的R_N-轨道切空间及R_N-无限决定性

本文研究了光滑映射芽在R_N作用下的轨道切空间的问题.利用乘积积分理论的方法,获得了光滑映射芽关于右等价群的一类子群R_N的无限决定性的结果,推广了光滑映射芽是R_N-有限决定的结果.     

在左右等价下余维数不大于3的Z_4-不变势函数芽的分类

以紧致Lie群Z_4为对称群,讨论在左右等价群下Z_4-不变势函数芽的分类问题.分别给出了Z_4和D_4-不变函数芽环的Hilbert基,得到了Z_4-不变函数芽环可以看成是D_4-不变函数芽环上的有限生成模的结论.通过将D_4-不变函数芽环复化,将Z_4-等变映射芽模看成该复化环上的有限生成模.因此将Z_4-不变势函数芽的分类问题转化成D_4-不变函数芽环上的有限生成模的讨论.给出了一定非退化条件下余维数不大于3的Z_4-不变函数芽的分类,并得到了相应的标准形式.     

光滑映射芽的R_k—有限决定性

本文刻划了光滑映射芽是R_k—有限决定的特征,并且对决定性阶数进行了估计。文中的诸结论在实际运用中主要用于光滑函数芽。     

c~∞函数有限决定性的充要条件

在c~∞函数的奇点理论中,通常把c~∞函数局部地分做所谓微分芽的等价类,J.Mather证明了对微分芽的有限决定性,有一个充分条件和一个必要条件.即(1)若m(n)~k m(n)+m(n)~(k+1),则f是k决定的.(2)若f是k决定的,则m(n)~(k+1) m(n).D.Siersma证明,对齐次多项式来说,(2)也是充分的.M.A.B.,Dekin把c~∞函数分成更粗的等价类.f,g∈c~∞(n,1)叫做T等价的,若它们在原点x=0的Taylor级数完全相同,此即形式幂级数的方法.     

等变序列分歧问题关于右等价群的有限决定性

研究在右等价群R(τ)的作用下,等变序列分歧问题f∈ε_(uλ)×(?)(τ)的有限决定性,得到了有限决定性的充分条件和必要条件.     

映射芽的V_q决定性

本文给出判别映射芽有限决定的一个代数条件和映射芽的?决定性和V_q决定性的关系。     

带参数的函数芽的无限决定性

本文讨论奇点理论在分问题中的应用．用有限决定性的方法于光滑函数的奇点，得到如下结果：当m(n q)^∞包含m(n q)（δf/δx)时，带参数的函数芽f是无穷决定的．这一结果发展了Wilson定理([2])，可应用于分支问题．     

等变分歧问题的无穷小稳定开拓

基于奇点理论中光滑函数芽的左右等价,本文讨论等变分歧问题开折的稳定性,刻画了有限型等变分歧问题的无穷小开折的稳定性,并指出这类分歧问题A(Γ)-通用开折必为无穷小稳定开折.     

Formal and finite order equivalences

We show that two families of germs of real-analytic subsets in ${{\mathbb C}^{n}}$ are formally equivalent if and only if they are equivalent of any finite order. We further apply the same technique to obtain analogous statements for equivalences of real-analytic self-maps and vector fields under conjugations. On the other hand, we provide an example of two sets of germs of smooth curves that are equivalent of any finite order but not formally equivalent.     

Reflection Ideals and mappings between generic submanifolds in complex space

Results on finite determination and convergence of formal mappings between smooth generic submanifolds in ℂ ^N are established in this article. The finite determination result gibes sufficient conditions to guarantee that a formal map is uniquely determined by its jet, of a preassigned order, at a point. Convergence of formal mappings for real-analytic generic submanifolds under appropriate assumptions is proved, and natural geometric conditions are given to assure that if two germs of such submanifolds are formally equivalent, then, they are necessarily biholomorphically equivalent. It is also shown that if two real-algebraic hypersurfaces in ℂ ^N are biholomorphically equivalent, then, they are algebraically equivalent. All the results are first proved in the more general context of “reflection ideals” associated to formal mappings between formal as well as real-analytic and real-algebraic manifolds.     

Finite Determinacy of High Codimension Smooth Function Germs

Mather gave the necessary and suffcient conditions for the ?nite determinacy smooth function germs with no more than codimension 4. The theorem is very effective on determining low codimension smooth function germs. In this paper, the concept of right equivalent for smooth function germs ring generated by two ideals ?nitely is de?ned. The containment relationships of function germs still satisfy ?nite k-determinacy under suffciently small disturbance which are discussed in orbit tangent spaces. Furthermore, the methods in judging the right equivalency of Arnold function family with codimension 5 are presented.     

The Poincaré series of divisorial valuations in the plane defines the topology of the set of divisors

With a plane curve singularity one associates a multi-index filtration on the ring of germs of functions of two variables defined by the orders of a function on irreducible components of the curve. The Poincaré series of this filtration turns out to coincide with the Alexander polynomial of the curve germ. For a finite set of divisorial valuations on the ring corresponding to some components of the exceptional divisor of a modification of the plane, in a previous paper there was obtained a formula for the Poincaré series of the corresponding multi-index filtration similar to the one associated with plane germs. Here we show that the Poincaré series of a set of divisorial valuations on the ring of germs of functions of two variables defines “the topology of the set of the divisors” in the sense that it defines the minimal resolution of this set up to combinatorial equivalence. For the plane curve singularity case, we also give a somewhat simpler proof of the statement by Yamamoto which shows that the Alexander polynomial is equivalent to the embedded topology.     

P_n(Γ)一组Hilbert基的判定和计算

设Γ是一作用在Rn上的紧李群,Pn(Γ)是Γ不变的多项式芽环,Hilbert-Weyl定理证明了对于Pn(Γ)总存在一组由Γ不变的齐次多项式芽构成的Hilbert基.然而,如何从Γ不变的齐次多项式芽中选出一组Hilbert基?如何判定Γ不变的齐次多项式芽的一个有限集就是Pn(Γ)的一组Hilbert基?在有关的文献中,Pn(Γ)的一组Hilbert基常常是通过幂级数展开进行计算.作为一个补充,本文借助Noether环、不变积分的基本性质以及奇点理论的某些定理,证明了判定、计算Pn(Γ)的Hilbert基的有关定理和原理,这提供了计算某些Pn(Γ)一组Hilbert基的而与幂级数展开不同的方法.最后,举例加以说明.     

关于映射芽在A和K的一些子群下的有限决定性

本文利用乘积积分理论给出了映射芽在Ａ和Ｋ的一些子群下有限决定的充分必要条件     

Hilbert function,generalized Poincaré series and topology of plane valuations

To a multi-index filtration (say, on the ring of germs of functions on a germ of a complex analytic variety) one associates several invariants: the Hilbert function, the Poincaré series, the generalized Poincaré series, and the generalized semigroup Poincaré series. The Hilbert function and the generalized Poincaré series are equivalent in the sense that each of them determines the other one. We show that for a filtration on the ring of germs of holomorphic functions in two variables defined by a collection of plane valuations both of them are equivalent to the generalized semigroup Poincaré series and determine the topology of the collection of valuations, i.e. the topology of its minimal resolution.     

Most real analytic Cauchy-Riemann manifolds are nonalgebraizable

We give a simple argument to the effect that most germs of generic real analytic Cauchy-Riemann manifolds of positive CR dimension are not holomorphically embeddable into a generic real algebraic CR manifold of the same real codimension in a finite dimensional space. In particular, most such germs are not holomorphically equivalent to a germ of a generic real algebraic CR manifold.Mathematics Subject Classification (2000): Primary 32V20, 32V30Supported in part by Research Program P1-0291, Republic of SloveniaAcknowledgement I wish to thank Peter Ebenfelt and Alexander Sukhov for their invaluable advice concerning the state of knowledge on the question considered in the paper.     

判定P_n(Γ)的Hilbert基的一个充要

设Γ是一作用在R^n上的紧李群，P_n(Γ)是Γ不变的多项式芽构成的环. Hilbert-Weyl定理证明了对于P_n(Γ)总存在一组由Γ不变的齐次多项式芽组成的Hilbert基. 然而，如何从Γ不变的齐次多项式芽中选出一组Hilbert基？如何判定Γ不变的齐次多项式芽的一个有限集就是P_n(Γ)的一组Hilbert基？该文借助于Noether环和不变积分的某些基本性质以及奇点理论的有关定理，证明了判定P_n(Γ)的Hilbert基的一个充要条件. 这对某些P_n(Γ)提供了计算一组Hilbert基的新途径.