Papers › Roots multiplicity and square-free factorization of polynomials using companion matrices
Roots multiplicity and square-free factorization of polynomials using companion matrices
Natalio H. Guersenzvaig, Fernando Szechtman
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Given an arbitrary monic polynomial f over a field F of characteristic 0, we use companion matrices to construct a polynomial M_f∈F[X] of minimum degree such that for each root α of f in the algebraic closure of F, M_f(α) is equal to the multiplicity m(α) of α 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=P₁P₂²⋯Pₘᵐ, where Pₖ is the product of all X-α with m(α)=k, for k=1, …, m=maxm(α).
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