多项式系统实根的连分式展开
Continued Fraction Expansion of Real Roots of Polynomial Systems
摘要：
本文提出了一种新的算法，用于隔离多元多项式系统的实根，在单项式基础上。它的灵感来自于现有的细分方法在伯恩斯坦基础;它可以被看作是推广的单变量连分式算法或替代作为一个完全模拟伯恩斯坦细分在单项基础。细分域的表示是通过单应性完成的，这使得我们可以只使用整数运算，并有效地处理无界区域。我们使用单变量边界函数，投影和预处理技术，以减少搜索的域。由此产生的框具有优化的有理坐标，对应于实根的连分式展开的第一项。文生定理的多元多项式的一个扩展被证明和用于终止的算法。新的复杂性界限提供了一个简化版本的算法。初步的C++实现计算的例子说明了这种方法。
英文摘要：
We present a new algorithm for isolating the real roots of a system of multivariate polynomials, given in the monomial basis. It is inspired by existing subdivision methods in the Bernstein basis; it can be seen as generalization of the univariate continued fraction algorithm or alternatively as a fully analog of Bernstein subdivision in the monomial basis. The representation of the subdivided domains is done through homographies, which allows us to use only integer arithmetic and to treat efficiently unbounded regions. We use univariate bounding functions, projection and preconditionning techniques to reduce the domain of search. The resulting boxes have optimized rational coordinates, corresponding to the first terms of the continued fraction expansion of the real roots. An extension of Vincent's theorem to multivariate polynomials is proved and used for the termination of the algorithm. New complexity bounds are provided for a simplified version of the algorithm. Examples computed with a preliminary C++ implementation illustrate the approach.
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