A conical approach to Laurent expansions for multivariate meromorphic germs with linear poles
| Autor: | Li Guo, Sylvie Paycha, Bin Zhang |
|---|---|
| Rok vydání: | 2020 |
| Předmět: |
Pure mathematics
General Mathematics Laurent series Holomorphic function 01 natural sciences Exponential integral Mathematics::Algebraic Geometry Mathematics - Metric Geometry 0103 physical sciences FOS: Mathematics Mathematics - Combinatorics Uniqueness Complex Variables (math.CV) 0101 mathematics Linear combination 32A20 32A27 52A20 52C07 Meromorphic function Mathematics Mathematics - Complex Variables Mathematics::Complex Variables 010102 general mathematics Regular polygon Metric Geometry (math.MG) Combinatorics (math.CO) 010307 mathematical physics Algebraic fraction |
| Zdroj: | Pacific Journal of Mathematics. 307:159-196 |
| ISSN: | 1945-5844 0030-8730 |
| DOI: | 10.2140/pjm.2020.307.159 |
| Popis: | We use convex polyhedral cones to study a large class of multivariate meromorphic germs, namely those 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 non-holomorphic 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 non-holomorphicity of linear combinations of polar germs. This yields the uniqueness of the above sum when required to be supported on a suitable family of cones and assigns a Laurent expansion to the germ. Laurent expansions provide various decompositions of such germs and thereby a uniformized proof of known results on decompositions of rational fractions. These Laurent expansions also yield new concepts on the space of such germs, all of which are independent of the choice of the specific Laurent expansion. These include a generalization of Jeffrey-Kirwan's residue, a filtered residue and a coproduct in the space of such germs. When applied to exponential sums on rational convex polyhedral cones, the filtered residue yields back exponential integrals. 30 pages |
| Databáze: | OpenAIRE |
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