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arXiv · 1306.4342

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

Natalio H. Guersenzvaig·Fernando Szechtman
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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 $α$ 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 $f=P_1P_2^2\cdots P_m^m$, where $P_k$ is the product of all $X-α$ with $m(α)=k$, for $k=1, \dots, m=\max m(α)$.

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Natalio H. Guersenzvaig, Fernando Szechtman. 2013-06-18. Roots multiplicity and square-free factorization of polynomials using companion matrices. https://arxiv.org/abs/1306.4342

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