Decomposing Polynomial Sets Simultaneously into Gröbner Bases and Normal Triangular Sets

Characteristic decomposition of polynomial sets

“…需要特别指出的是, 该 算法假设序条件总满足, 但这一假设并非总是成立. 为了处理序条件不满足的情形, 文献 [46] 提出了 临时换序策略, 寻找可以让序条件满足的变元序以便可以继续进行分裂. 输入: 本小节中所提到的算法的详细伪代码描述以及正确性和终止性证明可参见文献 [48].…”

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

“…regular chains, normal chains and square-free triangular sets. And, lots of triangular-decomposition algorithms have been proposed, see for example [1,6,7,12,13,19,20,[22][23][24][25]34]. Nevertheless, classical triangular-decomposition algorithms are mainly based on factorization and pseudo-division and are well known not so efficient on big examples.…”

“…Wang proved that if the variable ordering condition is satisfied for a W-characteristic set, then the regularity and normality of the W-characteristic set are equivalent. Later, Dong and Mou proposed some algorithms in [12,13] for characteristic decomposition which is also a type of triangular decomposition consisting of normal chains. Owe to efficient computation of Gröbner bases, their algorithms perform better on some complicated polynomial systems than the classical algorithms.…”