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Triangular decomposition with different properties has been used for varioustypes of problem solving, e.g. geometry theorem proving, real solutionisolation of zero-dimensional polynomial systems, etc. In this paper, theconcepts of strong chain and square-free strong triangular decomposition(SFSTD) of zero-dimensional polynomial systems are defined. Because of its goodproperties, SFSTD may be a key way to many problems related to zero-dimensionalpolynomial systems, such as real solution isolation and computing radicals ofzero-dimensional ideals. Inspired by the work of Wang and of Dong and Mou, wepropose an algorithm for computing SFSTD based on Gr\"obner bases computation.The novelty of the algorithm is that we make use of saturated ideals andseparant to ensure that the zero sets of any two strong chains have nointersection and every strong chain is square-free, respectively. On one hand,we prove that the arithmetic complexity of the new algorithm can be singleexponential in the square of the number of variables, which seems to be amongthe rare complexity analysis results for triangular-decomposition methods. Onthe other hand, we show experimentally that, on a large number of examples inthe literature, the new algorithm is far more efficient than a populartriangular-decomposition method based on pseudo-division. Furthermore, it isalso shown that, on those examples, the methods based on SFSTD for realsolution isolation and for computing radicals of zero-dimensional ideals arevery efficient.

Square-free Strong Triangular Decomposition of Zero-dimensional Polynomial Systems