Isolating clusters of zeros of analytic systems using arbitrary-degree inflation

TL;DR

This paper introduces a method called inflation to isolate and enclose clusters of zeros of analytic systems within specific regions, improving zero localization techniques.

Contribution

It develops a novel inflation-based approach that relates the original system to a more manageable one, enabling precise zero cluster isolation.

Findings

Successfully constructs regions containing zero clusters without other zeros.

Provides a method to separate a zero cluster from the rest of the zeros.

Enhances the accuracy of zero localization in analytic systems.

Abstract

Given a system of analytic functions and an approximation to a cluster of zeros, we wish to construct two regions containing the cluster and no other zeros of the system. The smaller region tightly contains the cluster while the larger region separates it from the other zeros of the system. We achieve this using the method of inflation which, counterintuitively, relates it to another system that is more amenable to our task but whose associated cluster of zeros is larger.