# A Characterization of Multiplicity-Preserving Global Bifurcations of Complex Polynomial Vector Fields

## TL;DR

This paper characterizes how complex polynomial vector fields undergo bifurcations that preserve equilibrium multiplicities, showing they can be decomposed into simpler, well-understood bifurcations.

## Contribution

It provides a detailed characterization and decomposition of multiplicity-preserving bifurcations in complex polynomial vector fields of arbitrary degree.

## Key findings

- Any such bifurcation can be realized as a composition of simpler bifurcations.
- The simpler bifurcations involved are explicitly characterized.
- The results apply to polynomial vector fields of any degree d.

## Abstract

For the space of single-variable monic and centered complex polynomial vector fields of arbitrary degree d, it is proved that any bifurcation which preserves the multiplicity of equilibrium points can be realized as a composition of a finite number of simpler bifurcations, and these bifurcations are characterized.