Abstract
It is well known that, in the plane, the boundary of any quadrature domain (in the classical sense) coincides with the zero set of a polynomial. We show, by explicitly constructing some four-dimensional examples, that this is not always the case. This confirms, in dimension 4, a conjecture of the second author. Our method is based on the Schwarz potential and involves elliptic integrals of the third kind.
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Eremenko, A., Lundberg, E. Non-Algebraic Quadrature Domains. Potential Anal 38, 787–804 (2013). https://doi.org/10.1007/s11118-012-9297-6
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DOI: https://doi.org/10.1007/s11118-012-9297-6