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

Abstract

Let F be a field of characteristic 0 and let f be a monic polynomial of positive degree in F[X]. Let K be a splitting field for f over F and let α1,…,αs be the distinct roots of f in K with respective multiplicities m(α1),…,m(αs). By Lagrange Interpolation Formula there is a unique polynomial Mf∈K[X] of degree less than s satisfying Mf(αj)=m(αj) for 1⩽j⩽s. The polynomial Mf actually lies in F[X] and we present an explicit rational procedure to obtain Mf from f by means of companion matrices. As an application of Mf, we furnish a new method to compute each component of the square-free factorization f=P1P22⋯Pmm, where m=max1⩽j⩽sm(αj) and Pk is the product of all X-αj such that m(αj)=k