Germs of multivariate meromorphic functions with linear poles at zero> naturally arise in various contexts,

in number theory (multizeta functions), the combinatorics on cones (exponential sums) and also in quantum

field theory (Feynman diagrams). We want to extend to multivariate germs, concepts and results known for

meromorphic germs in one variable, such as Laurent expansions and the residue. Going from one to several

variables introduces new difficulties, which we tackle using the geometry of cones in an essential manner.

We discuss> various applications of these results, revisiting Berline and Vergne's Euler-Maclaurin formula

on cones and generalising the Jeffrey-Kirwan residue.

This is based on joint work with Li Guo and Bin Zhang.