Solving Zero-Dimensional Systems through the Rational Univariate Representation
Résumé
This paper is devoted to the 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 follow the definition basically due to Kronecker for solving zero-dimensional systems : A system is solved if each root is represented in such way as to allow the performance of any arithmetical operations over the arithmetical expressions of its coordinates. We propose new definitions for solving zero-dimensional systems in this sense by introducing the 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 (Rational Univariate Representation): $$ \{ f(T)=0 \;,\; X_1=\frac{g_1(T)}{g(T)}\;,\; \ldots \;,\; X_n=\frac{g_n(T)}{g(T)} \} $$ where $(f,g,g_1,\ldots ,g_n)$ are polynomials of $K[X_1,\ldots ,X_n]$. A special feature of our Rational Univariate Representation is that we don't loose geometrical information contained in the initial system. Moreover we propose different efficient algorithms for the computation of the Rational Univariate Representation, and we make a comparison with standard known tools.