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

2013/06/18 by Natalio H. Guersenzvaig, Guersenzvaig, Natalio H., Fernando Szechtman +1
Computer Science · Mathematics · #12D05 #13A05 #15A24 #Advanced Optimization Algorithms Research #FOS: Mathematics #Iterative Methods for Nonlinear Equations #Matrix Theory and Algorithms #Rings and Algebras (math.RA)

paper · pdf · doi:10.48550/arxiv.1306.4342

openalex publication_date 2013/06/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Given an arbitrary monic polynomial f over a field F of characteristic 0,\nwe use companion matrices to construct a polynomial M_f\∈ F[X] of minimum\ndegree such that for each root \α of f in the algebraic closure of F,\nM_f(\α) is equal to the multiplicity m(\α) of \α as a root of\nf. As an application of M_f we give a new method to compute in F[X] each\ncomponent of the square-free factorization f=P₁P₂²\⋯ P_m^m, where\nP_k is the product of all X-\α with m(\α)=k, for k=1, \…,\nm=\max m(\α).\n

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