#### Vol. 307, No. 1, 2020

 Recent Issues Vol. 307: 1  2 Vol. 306: 1  2 Vol. 305: 1  2 Vol. 304: 1  2 Vol. 303: 1  2 Vol. 302: 1  2 Vol. 301: 1  2 Vol. 300: 1  2 Online Archive Volume: Issue:
 The Journal Editorial Board Subscriptions Officers Special Issues Submission Guidelines Submission Form Contacts ISSN: 1945-5844 (e-only) ISSN: 0030-8730 (print) Author Index To Appear Other MSP Journals
A conical approach to Laurent expansions for multivariate meromorphic germs with linear poles

### Li Guo, Sylvie Paycha and Bin Zhang

Vol. 307 (2020), No. 1, 159–196
##### Abstract

We develop a geometric approach using convex polyhedral cones to build Laurent expansions for multivariate meromorphic germs with linear poles, which naturally arise in various contexts in mathematics and physics. We express such a germ as a sum of a holomorphic germ and a linear combination of special nonholomorphic germs called polar germs. In analyzing the supporting cones — cones that reflect the pole structure of the polar germs — we obtain a geometric criterion for the nonholomorphicity of linear combinations of polar germs. For any given germ, the above decomposition yields a Laurent expansion which is unique up to suitable subdivisions of the supporting cones. These Laurent expansions lead to new concepts on the space of meromorphic germs, such as a generalization of Jeffrey–Kirwan’s residue and a filtered residue, all of which are independent of the choice of the specific Laurent expansion.

##### Keywords
meromorphic function, convex cone, Laurent expansion, residue, Jeffrey–Kirwan residue
##### Mathematical Subject Classification 2010
Primary: 32A20, 32A27, 52A20, 52C07