Solving Zero-dimensional Polynomial Systems through the Rational Univariate Representation
Résumé
This paper is devoted to the {\it resolution} of zero-dimensional systems in $K[X_1,\ldots X_n]$, where $K$ is a field of characteristic zero (or strictly positive under some conditions). We give a new definition for {\rm solving zero-dimensional systems} by introducing the {\it Univariate Representation} of their roots. We show by this way that the solutions of any zero-dimensional system of polynomials can be expressed through a special kind of univariate representation ({\it Rational Univariate Representation}): $$ \begin{array}{c} f(T)=0 \\ X_1=\frac{g_1(T)}{g(T)} \\ \vdots \\ X_n=\frac{g_n(T)}{g(T)} \\ \end{array} $$ where $(f,g,g_1,\ldots ,g_n)$ are polynomials of $K[X_1,\ldots ,X_n]$, without loosing geometrical information (multiplicities, real roots). Moreover we propose different efficient algorithms for the computation of the {\it Rational Univariate Representation}, and we make a comparison with standard known tools.