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On the Exponential Transform of Multi-Sheeted Algebraic Domains

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We introduce multi-sheeted versions of algebraic domains and quadrature domains, allowing them to be branched covering surfaces over the Riemann sphere. The two classes of domains turn out to be the same, and the main result states that the extended exponential transform of such a domain agrees, apart from some simple factors, with the extended elimination function for a generating pair of functions. In an example we discuss the algebraic curves associated to level curves of the Neumann oval, and determine which of these give rise to multi-sheeted algebraic domains.

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Correspondence to Björn Gustafsson.

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This research has been supported by the Swedish Research Council, the Göran Gustafsson Stiftelse, and is part of the European Science Foundation Networking programme “Harmonic and Complex Analysis and Applications HCAA”.

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Gustafsson, B., Tkachev, V.G. On the Exponential Transform of Multi-Sheeted Algebraic Domains. Comput. Methods Funct. Theory 11, 591–615 (2012).

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2000 MSC