Deaconstructing Functions on Quadratic Surfaces into Multipoles

Abstract

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.