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Let $f$ be a polynomial system consisting of $n$ polynomials $f_1,\cdots,f_n$ in $n$ variables $x_1,\cdots, x_n$, with coefficients in $\mathbb{Q}$ andlet $\langle f\rangle$ be the ideal generated by $f$. Such a polynomial system,which has as many equations as variables is called a square system. It may bezero-dimensional, i.e the system of equations $f = 0$ has finitely many complexsolutions, or equivalently the dimension of the quotient algebra $A =\mathbb{Q}[x]/\langle f\rangle$ is finite. In this case, the companion matricesof $f$ are defined as the matrices of the endomorphisms of $A$, calledmultiplication maps, $x_j : \left\vert \begin{array}{c} h \mapsto x_jh\end{array} \right.$, written in some basis of $A$. We present a practical andefficient method to compute the companion matrices of $f$ in the case when thesystem is zero-dimensional. When it is not zero-dimensional, then the methodworks as well and still produces matrices having properties similar to thezero-dimensional case. The whole method consists in matrix calculations. Anexperiment illustrates the method's effectiveness.

Solving square polynomial systems : a practical method using Bezout matrices