Solving square polynomial systems : a practical method using Bezout matrices

Abstract

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 $\mathbb{Q}$ and let $\langle f\rangle$ be the ideal generated by $f$. Such a polynomial system, which has as many equations as variables is called a square system. It may be zero-dimensional, i.e the system of equations $f = 0$ has finitely many complex solutions, or equivalently the dimension of the quotient algebra $A = \mathbb{Q}[x]/\langle f\rangle$ is finite. In this case, the companion matrices of $f$ are defined as the matrices of the endomorphisms of $A$, called multiplication 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 and efficient method to compute the companion matrices of $f$ in the case when the system is zero-dimensional. When it is not zero-dimensional, then the method works as well and still produces matrices having properties similar to the zero-dimensional case. The whole method consists in matrix calculations. An experiment illustrates the method's effectiveness.