# The secant map applied to a real polynomial with multiple roots

## TL;DR

This paper studies the behavior of the secant map when applied to polynomials with multiple roots, revealing how local dynamics depend on the roots' multiplicity parity.

## Contribution

It provides a detailed analysis of the local dynamics of the secant map near roots with multiple multiplicities, highlighting the influence of parity.

## Key findings

- Dynamics depend on the parity of the root multiplicity
- Fixed points' stability varies with multiplicity parity
- Provides insights into polynomial root behavior under secant iteration

## Abstract

We investigate the plane dynamical system given by the secant map applied to a polynomial $p$ having at least one multiple root of multiplicity $d>1$. We prove that the local dynamics around the fixed points associated to the roots of $p$ depend on the parity of $d$.