The main goal of this work is classifying the singularities of slice regularfunctions over a real alternative *-algebra A. This function theory has beenintroduced in 2011 as a higher-dimensional generalization of the classicaltheory of holomorphic complex functions, of the theory of slice regularquaternionic functions launched by Gentili and Struppa in 2006 and of thetheory of slice monogenic functions constructed by Colombo, Sabadini andStruppa since 2009. Along with this generalization step, the larger class ofslice functions over A has been defined. We introduce here a new type of seriesexpansion near each singularity of a slice regular function. This instrument,which is new even in the quaternionic case, leads to a complete classificationof singularities. This classification also relies on some recent developmentsof the theory, concerning the algebraic structure and the zero sets of slicefunctions. Peculiar phenomena arise, which were not present in the complex orquaternionic case, and they are studied by means of new results on the topologyof the zero sets of slice functions. The analogs of meromorphic functions,called (slice) semiregular functions, are introduced and studied.
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