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Abstract
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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.
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Keywords
meromorphic function, convex cone, Laurent expansion,
residue, Jeffrey–Kirwan residue
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Mathematical Subject Classification 2010
Primary: 32A20, 32A27, 52A20, 52C07
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Milestones
Received: 26 April 2019
Revised: 20 February 2020
Accepted: 23 February 2020
Published: 8 August 2020
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