摘要翻译:
本文考虑一个复变多项式映射的仿射共轭类族$\mathrm{MP}_d$,并研究了映射$\phi_d:\mathrm{MP}_d\到\widitilde{\lambda}_d\子集\mathbb{C}^d/\mathfrak{S}_d$,它将mathrm{MP}_d$中的每个$f\映射到$f$的不动点乘子集。我们证明了映射$\phi_d$在$\bar{\lambda}\in\widitilde{\lambda}_d$周围的局部光纤结构完全由两个集合$\mathcal{I}(\lambda)$和$\mathcal{K}(\lambda)$决定,这两个集合是$\1,2,\ldots,d\}$的幂集的子集。此外,对于任何$\bar{\lambda}\in\widetilde{\lambda}_d$,我们只使用$\mathcal{I}(\lambda)$和$\mathcal{K}(\lambda)$就给出了计算每个纤维$\phi_d^{-1}\left(\bar{\lambda}\right)$元素数的算法。它可以在有限的许多步骤中进行,而且通常是手工进行的。
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英文标题:
《The Moduli Space of Polynomial Maps and Their Fixed-Point Multipliers》
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作者:
Toshi Sugiyama
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最新提交年份:
2017
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分类信息:
一级分类:Mathematics 数学
二级分类:Algebraic Geometry 代数几何
分类描述:Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇,叠,束,格式,模空间,复几何,量子上同调
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一级分类:Mathematics 数学
二级分类:Complex Variables 复变数
分类描述:Holomorphic functions, automorphic group actions and forms, pseudoconvexity, complex geometry, analytic spaces, analytic sheaves
全纯函数,自守群作用与形式,伪凸性,复几何,解析空间,解析束
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一级分类:Mathematics 数学
二级分类:Dynamical Systems 动力系统
分类描述:Dynamics of differential equations and flows, mechanics, classical few-body problems, iterations, complex dynamics, delayed differential equations
微分方程和流动的动力学,力学,经典的少体问题,迭代,复杂动力学,延迟微分方程
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一级分类:Mathematics 数学
二级分类:Geometric Topology 几何拓扑
分类描述:Manifolds, orbifolds, polyhedra, cell complexes, foliations, geometric structures
流形,轨道,多面体,细胞复合体,叶状,几何结构
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英文摘要:
We consider the family $\mathrm{MP}_d$ of affine conjugacy classes of polynomial maps of one complex variable with degree $d \geq 2$, and study the map $\Phi_d:\mathrm{MP}_d\to \widetilde{\Lambda}_d \subset \mathbb{C}^d / \mathfrak{S}_d$ which maps each $f \in \mathrm{MP}_d$ to the set of fixed-point multipliers of $f$. We show that the local fiber structure of the map $\Phi_d$ around $\bar{\lambda} \in \widetilde{\Lambda}_d$ is completely determined by certain two sets $\mathcal{I}(\lambda)$ and $\mathcal{K}(\lambda)$ which are subsets of the power set of $\{1,2,\ldots,d \}$. Moreover for any $\bar{\lambda} \in \widetilde{\Lambda}_d$, we give an algorithm for counting the number of elements of each fiber $\Phi_d^{-1}\left(\bar{\lambda}\right)$ only by using $\mathcal{I}(\lambda)$ and $\mathcal{K}(\lambda)$. It can be carried out in finitely many steps, and often by hand.
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PDF链接:
https://arxiv.org/pdf/0708.2512