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| B. Gustafsson showed (1983) that planar quadrature domains are always algebraic (they have boundary contained in the zero set of a polynomial). H. S. Shapiro asked (1992) whether this is true in higher dimensions. We answer this question in the negative by constructing explicit four-dimensional examples that have transcendental boundary. In terms of exterior potential, our examples behave like a point charge superimposed on a dipole source. The construction involves elliptic integrals of the third kind and uses the relation between the Schwarz potential (in four dimensions) and the Schwarz function of the generating curve in the plane. This is joint work with Alexandre Eremenko. |
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