Roots multiplicity and square-free factorization of polynomials using companion matrices

TL;DR

This paper introduces a novel method using companion matrices to determine root multiplicities and perform square-free factorization of polynomials over fields of characteristic zero, enhancing algebraic computation techniques.

Contribution

The paper presents a new approach employing companion matrices to compute root multiplicities and square-free factors of polynomials in a more efficient and algebraically insightful manner.

Findings

Constructed a polynomial $M_f$ encoding root multiplicities.

Provided a new method for square-free factorization in $F[X]$.

Demonstrated the effectiveness of the approach through algebraic applications.

Abstract

Given an arbitrary monic polynomial $f$ over a field $F$ of characteristic 0, we use companion matrices to construct a polynomial $M_f\in F[X]$ of minimum degree such that for each root $\alpha$ of $f$ in the algebraic closure of $F$, $M_f(\alpha)$ is equal to the multiplicity $m(\alpha)$ of $\alpha$ as a root of $f$. As an application of $M_f$ we give a new method to compute in $F[X]$ each component of the square-free factorization $f=P_1P_2^2\cdots P_m^m$, where $P_k$ is the product of all $X-\alpha$ with $m(\alpha)=k$, for $k=1, \dots, m=\max m(\alpha)$.