从有限个广义极化张量识别二维代数域
Identification of an algebraic domain in two dimensions from a finite number of its generalized polarization tensors
摘要：
本文研究了用代数域的几个广义极化张量来确定其形状的问题。精确地说，给定一个具有实代数边界的平面集合，证明了实系数在边界上为零的极小多项式可以被识别为一个矩阵的一维核的生成元，该矩阵的元素由有限个广义极化张量得到。矩阵的大小多项式地依赖于代数域的边界的次数。在所有有界域之间的代数域的Hausdorff距离的密度邀请通过近似扩展我们的重建过程超出其自然的背景。在此基础上，提出了一种新的形状识别/分类算法，并对其效率进行了一些有力的暗示。
英文摘要：
This paper aims at studying how finitely many generalized polarization tensors of an algebraic domain can be used to determine its shape. Precisely, given a planar set with real algebraic boundary, it is shown that the minimal polynomial with real coefficients vanishing on the boundary can be identified as the generator of a one dimensional kernel of a matrix whose entries are obtained from a finite number of generalized polarization tensors. The size of the matrix depends polynomially on the degree of the boundary of the algebraic domain. The density with respect to Hausdorff distance of algebraic domains among all bounded domains invites to extend via approximation our reconstruction procedure beyond its natural context. Based on this, a new algorithm for shape recognition/classification is proposed with some strong hints about its efficiency.
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