Triangular decomposition with different properties has been used for various types of problem solving, e.g. geometry theorem proving, real solution isolation of zero-dimensional polynomial systems, etc. In this paper, the concepts of strong chain and square-free strong triangular decomposition (SFSTD) of zero-dimensional polynomial systems are defined. Because of its good properties, SFSTD may be a key way to many problems related to zero-dimensional polynomial systems, such as real solution isolation and computing radicals of zero-dimensional ideals. Inspired by the work of Wang and of Dong and Mou, we propose an algorithm for computing SFSTD based on Gr\"obner bases computation. The novelty of the algorithm is that we make use of saturated ideals and separant to ensure that the zero sets of any two strong chains have no intersection and every strong chain is square-free, respectively. On one hand, we prove that 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, we 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 pseudo-division. Furthermore, it is also shown that, on those examples, the methods based on SFSTD for real solution isolation and for computing radicals of zero-dimensional ideals are very efficient.
翻译:具有不同特性的三角分解已经用于各类解决问题的各类类型的问题,例如:几何定理理论的验证、零维多元海洋系统的真正溶解隔离,等等。在本文中,对零维多元海洋系统的强链和无平方强三角分解(SFSTD)概念进行了定义。由于其良好的特性,SFSTD可能是解决与零维多元纳米系统有关的许多问题的关键方法,例如真正的溶解隔离和计算零维理想的激进。在王、东和穆的工作启发下,我们提议了一个基于Gr\'obner基础计算来计算SFSTD的算法。这个算法的新颖概念是,我们使用饱和无方的三维多元多元多元多元多元多元多元多元多维系统(SFSTD)的零分解(SSFSTD)概念的概念,以确保任何两个强大链的零组合没有交叉,而每个强大链都是无正态的。一方面,我们证明新算算法的复杂性可以在变量的方形数中成为单一指数,这似乎是基于Grix-decom模型的罕见的复杂分析结果,我们所展示了基于大实验性模型的模型的其他方法。