Solving Zero-dimensional Polynomial Systems through the Rational Univariate Representation
Solving Zero-dimensional Polynomial Systems through the Rational Univariate Representation
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发表时间：
1998
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通讯作者：
F. Rouillier
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作者：
F. Rouillier
本文研究了零维系统在$K[X_1，X_n]中的分解，其中$K是特征为零的域(或在一定条件下是严格正的)。通过引入零维系统根的单变量表示，给出了零维系统解的一个新定义。通过这种方法，我们证明了任何零维多项式的解都可以用一种特殊的单变量表示(有理单变量表示)来表示：$$\Begin{ARRAY}{c}f(T)=0\\X_1=\FRAC{g_1(T)}{g(T)}\\Vdots\\X_n=\FRAC{g_n(T)}{g(T)}\\end{ARRAY}$其中$(f，g，g_1，\ldots，G_n)$是$K[X_1，\ldots，X_n]$的多项式，不丢失几何信息(重数，实根)。此外，我们还提出了计算有理单变量表示的不同有效算法，并与标准的已知工具进行了比较。
This paper is devoted to the {\it resolution} of zero-dimensional systems in $K[X_1,\ldots X_n]$, where $K$ is a field of characteristic zero (or strictly positive under some conditions). We give a new definition for {\rm solving zero-dimensional systems} by introducing the {\it Univariate Representation} of their roots. We show by this way that the solutions of any zero-dimensional system of polynomials can be expressed through a special kind of univariate representation ({\it Rational Univariate Representation}): $$ \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.