关于调和矩为零的多边形测度
On polygonal measures with vanishing harmonic moments
摘要：
一个带符号多边形测度是多边形上支持的2个实常数密度测度之和。给定平面上的一个有限集S，研究了顶点在S中的多边形所张成的调和矩为零的符号多边形测度的存在性。对于S-泛，我们证明了这类测度的线性空间的维数是（|S| -3）（|S| -4）/2.我们还调查的情况下，所得的密度是0，或1，或-1，这对应于对多边形的单位密度具有相同的对数电位在无穷大。证明了当|S|至多为5时，不存在这样一个带符号测度，但当n至少为6时，存在一个S，且|S| =n，从而产生这样一个带号测度.
英文摘要：
A signed polygonal measure is the sum of finitely many real constant density measures supported on polygons. Given a finite set S in the plane, we study the existence of signed polygonal measures spanned by polygons with vertices in S, which have all harmonic moments vanishing. For S generic, we show that the dimension of the linear space of such measures is (|S|-3)(|S|-4)/2. We also investigate the situation where the resulting density is either 0, or 1, or -1, which corresponds to pairs of polygons of unit density having the same logarithmic potential at infinity. We show that such a signed measure does not exist if |S| is at most 5, but for each n at least 6 there exists an S, with |S|=n, giving rise to such a signed measure.
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