Singularities of slice regular functions over real alternative ^⁎-algebras

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• 作者：Riccardo Ghiloni^a ; ^{riccardo.ghiloni@unitn.it" class="auth_mail" title="E-mail the corresponding author} ; Alessandro Perotti^a ; ^{alessandro.perotti@unitn.it" class="auth_mail" title="E-mail the corresponding author} ; Caterina Stoppato^b ; ^{stoppato@math.unifi.it" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Slice regular functions ; Quaternions ; Alternative *-algebras ; Power series and expansions ; Laurent series and expansions ; Spherical series and expansions ; Singularities ; Semiregular functions
• 刊名：Advances in Mathematics
• 出版年：2017
• 出版时间：10 January 2017
• 年：2017
• 卷：305
• 期：Complete
• 页码：1085-1130
• 全文大小：686 K

文摘

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