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摘要翻译：
任何次为$D$的齐次多项式$P(x，y，z)$,被限制在一个单位球面$S^2$内，本质上允许一个形式为$\lambda+\sum_{k=1}^d[\prod_{j=1}^k L_{kj}]$的唯一表示，其中$L_{kj}$是在$x，y$和$z$中的线性形式，$\lambda$是实数。这些线性形式的系数，看作三维向量，称为$p$的\emph{多极子}向量。本文讨论了其它二次曲面q(x，y，z)=c，实曲面和复曲面上多项式函数和解析函数的相似多极表示。在复数上，上面的表示不是唯一的，尽管歧义本质上是有限的。我们研究描述这种歧义的组合学。我们将这些结果与调和分析的一些经典定理联系起来，这些定理描述函数分解为球面调和和。我们将这些经典定理（依赖于我们对Laplace算子$\delta_{S^2}的理解）推广到借助二次型$q(x,y,z)$构造的更一般的微分算子$\delta_q$。然后我们引入了多极子的模空间。我们用代数几何和奇点理论的方法研究了它们复杂的几何和拓扑。多极空间是向量空间或射影空间上的分支，对分支集的补充得到了一个丰富的$k(\pi，1)$-空间族，其中$\pi$运行在各种修正辫子群上。
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英文标题：
《Deaconstructing Functions on Quadratic Surfaces into Multipoles》
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作者：
Gabriel Katz
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最新提交年份：
2007
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分类信息：
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：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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英文摘要：
Any homogeneous polynomial $P(x, y, z)$ of degree $d$, being restricted to a unit sphere $S^2$, admits essentially a unique representation of the form $\lambda + \sum_{k = 1}^d [\prod_{j = 1}^k L_{kj}]$, where $L_{kj}$'s are linear forms in $x, y$ and $z$ and $\lambda$ is a real number. The coefficients of these linear forms, viewed as 3D vectors, are called \emph{multipole} vectors of $P$. In this paper we consider similar multipole representations of polynomial and analytic functions on other quadratic surfaces $Q(x, y, z) = c$, real and complex. Over the complex numbers, the above representation is not unique, although the ambiguity is essentially finite. We investigate the combinatorics that depicts this ambiguity. We link these results with some classical theorems of harmonic analysis, theorems that describe decompositions of functions into sums of spherical harmonics. We extend these classical theorems (which rely on our understanding of the Laplace operator $\Delta_{S^2}$) to more general differential operators $\Delta_Q$ that are constructed with the help of the quadratic form $Q(x, y, z)$. Then we introduce modular spaces of multipoles. We study their intricate geometry and topology using methods of algebraic geometry and singularity theory. The multipole spaces are ramified over vector or projective spaces, and the compliments to the ramification sets give rise to a rich family of $K(\pi, 1)$-spaces, where $\pi$ runs over a variety of modified braid groups.
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PDF链接：
https://arxiv.org/pdf/0704.1174
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