Triangular decomposition with different properties has been used for various types of problem solving. In this paper, the concepts of pure chains and square-free pure triangular decomposition (SFPTD) of zero-dimensional polynomial systems are defined. Because of its good properties, SFPTD may be a key way to many problems related to zero-dimensional polynomial systems. Inspired by the work of Wang (2016) and of Dong and Mou (2019), the authors propose an algorithm for computing SFPTD based on Gröbner bases computation. The novelty of the algorithm is that the authors make use of saturated ideals and separant to ensure that the zero sets of any two pure chains are disjoint and every pure chain is square-free, respectively. On one hand, the authors prove the arithmetic complexity of the new algorithm can be single exponential in the square of the number of variables, which seems to be among the rare complexity analysis results for triangular-decomposition methods. On the other hand, the authors show experimentally that, on a large number of examples in the literature, the new algorithm is far more efficient than a popular triangular-decomposition method based on pseudodivision, and the methods based on SFPTD for real solution isolation and for computing radicals of zero-dimensional ideals are very efficient.

具有不同性质的三角分解已被用于各种类型的问题解决。在本文中，定义了零维多项式系统的纯链和无平方纯三角分解（SFPTD）的概念。由于其良好的性质，SFPTD 可能是解决与零维多项式系统相关的许多问题的关键方法。受 Wang (2016) 和 Dong 和 Mou (2019) 工作的启发，作者提出了一种基于 Gröbner 基计算的 SFPTD 计算算法。该算法的新颖之处在于作者分别利用饱和理想和分离性来保证任意两条纯链的零集不相交，且每条纯链都是无平方的。一方面，作者证明了新算法的算术复杂度可以是变量个数平方的单指数，这似乎是三角分解方法中罕见的复杂度分析结果之一。另一方面，作者通过实验表明，在文献中的大量实例上，新算法的效率远高于流行的基于伪分割的三角分解方法，以及基于 SFPTD 的实解分离和求解方法。对于计算零维理想的根是非常有效的。