Algebraicity of Quadrature Domains: 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. H. S. Shapiro and B. Gustafsson asked whether this is also the case in higher dimensions. Alex Eremenko and I showed that the answer is "not always" by explicitly constructing some four-dimensional examples that have transcendental boundary. Our method was based on the Schwarz potential and involves elliptic integrals of the third kind.

This confirmed, in dimension 4, a conjecture from my thesis. I expect that a stronger form of the statement is true:

In all dimensions greater than 2, quadrature domains are almost never algebraic, where "almost never" is in the strongest possible sense (in the parameter space of quadrature domains, the codimension is smaller than the dimension of non-algebraic examples).

For additional questions regarding higher-dimensional quadrature domains that would require other types of explicit examples, our methods can likely be used to construct them explicitly.

Open Problem: In two dimensions, there are examples of non-uniqueness due to Strakhov of unbounded one-point quadrature domains. Namely, there is a one-parameter family of domains all having the same quadrature formula. Find an explicit example in four dimensions.

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