Decomposition of polynomial sets into characteristic pairs

Abstract

A characteristic pair is a pair (G, C) of polynomial sets in which G is a reduced lexicographic Gröbner basis, C is the minimal triangular set contained in G, and C is normal. In this paper, we show that any finite polynomial set P can be decomposed algorithmically into finitely many characteristic pairs with associated zero relations, which provide representations for the zero set of P in terms of those of Gröbner bases and those of triangular sets. The algorithm we propose for the decomposition makes use of the inherent connection between Ritt characteristic sets and lexicographic Gröbner bases and is based essentially on the structural properties and the computation of lexicographic Gröbner bases. Several nice properties about the decomposition and the resulting characteristic pairs, in particular relationships between the Gröbner basis and the triangular set in each pair, are established. Examples are given to illustrate the algorithm and some of the properties.

Comprehensive Characteristic Decomposition of Parametric Polynomial Systems

“…In particular, a characteristic decomposition is said to be normal (regular, or strong) if each characteristic pair it contains is normal (regular, or strong, respectively). Properties of characteristic decomposition and algorithms for computing regular, normal, and strong characteristic decompositions based on pseudo-divisibility of polynomials within reduced Gröbner bases and ideal saturation and quotient are presented in [12,13,35].…”

“…The whole procedure outlined above is described formally as Algorithm 2. In this algorithm, the subalgorithm NormalCharDec takes a polynomial set as input and returns its normal characteristic decompositions [13,35]. Proof.…”

“…A simple question that has been asked is where and how the two parallel lines are interconnected. One nontrivial answer to this question was given in 2016 by the last author who established an inherent connection between lexicographical Gröbner bases and Ritt characteristic sets [34], and since then an algorithmic theory of characteristic decomposition has been developed to bridge Gröbner bases and triangular decompositions [12,13,35]. The bridge is built up via the key concept of W-characteristic sets, which are minimal triangular sets contained in lexicographical Gröbner bases, and the connotation of this concept is enriched with a structure theorem for irregular Gröbner bases.…”

Characteristic decomposition of polynomial sets

“…特别地, Gröbner 基作为多项式理想的良好表示, 从中提取出来的正规列或正则列构成的集合以更简洁的形式携带着相 似的零点信息. 有关更多 (强) 正规特征分解和 (强) 正则特征分解的理论基础和良好性质, 可参见文 献 [44,45]. 特别地, 文献 [45] 证明了正规特征分解的如下结论.…”

“…根据 W 特征列中特定多项式之间的伪整除关系, 文献 [44,45] 基于定理 3.1 提出了正规特征分解 算法. 而受代数簇除法运算的启发, 文献 [48] 提出了另一种强正则特征分解算法, 该算法主要基于 W 特征列的饱和理想以及理想商运算的分解策略, 优点在于不需要考虑序条件.…”