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arXiv · 1209.4014

On polygonal measures with vanishing harmonic moments

Dmitrii Pasechnik·Boris Shapiro
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Abstract

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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Dmitrii Pasechnik, Boris Shapiro. 2013-04-11. On polygonal measures with vanishing harmonic moments. https://doi.org/10.1007/s11854-014-0021-x

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