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It is proved that quadrature domains are ubiquitous in a very strong sense inthe realm of smoothly bounded multiply connected domains in the plane. In fact,they are so dense that one might as well assume that any given smooth domainone is dealing with is a quadrature domain, and this allows access to a host ofstrong conditions on the classical kernel functions associated to the domain.Following this string of ideas leads to the discovery that the Bergman kernelcan be zipped down to a strikingly small data set. It is also proved that thekernel functions associated to a quadrature domain must be algebraic.Comment: 13 pages, to appear in Arkiv for matemati

Quadrature domains and kernel function zipping