A topological splitting of the space of meromorphic germs in several variables and continuous evaluators

TL;DR

This paper establishes a topological decomposition of the space of multivariable meromorphic germs into holomorphic and polar parts, enabling continuous evaluation at zero and generalizing previous algebraic decompositions.

Contribution

It introduces a topological splitting of meromorphic germs in several variables using Silva spaces, extending prior algebraic decompositions to a topological framework.

Findings

Provides a continuous evaluation at zero for meromorphic germs.

Generalizes previous algebraic decompositions to a topological setting.

Uses Silva spaces and inner products to achieve the decomposition.

Abstract

We prove a topological decomposition of the space of meromorphic germs at zero in several variables with prescribed linear poles as a sum of spaces of holomorphic and polar germs. Evaluating the resulting holomorphic projection at zero gives rise to a continuous evaluator (at zero) on the space of meromorphic germs in several variables. Our constructions are carried out in the framework of Silva spaces and use an inner product on the underlying space of variables. They generalise to several variables, the topological direct decomposition of meromorphic germs at zero as sums of holomorphic and polar germs previously derived by the first and third author and provide a topological refinement of a known algebraic decomposition of such spaces previously derived by the second author and collaborators.