A conical approach to Laurent expansions for multivariate meromorphic germs with linear poles

TL;DR

This paper introduces a geometric framework using convex polyhedral cones to analyze multivariate meromorphic germs with linear poles, leading to new decompositions, residues, and algebraic structures.

Contribution

It develops a conical approach to Laurent expansions for germs with linear poles, establishing uniqueness, and introduces new residue concepts and algebraic tools.

Findings

Provided a geometric criterion for non-holomorphicity of polar germs

Established the uniqueness of Laurent expansions supported on cones

Introduced generalized residues and coproduct structures

Abstract

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.