Computing strong regular characteristic pairs with Gröbner bases

Dong, Rina; Wang, Dongming

doi:10.1016/j.jsc.2020.06.012

Cited by 5 publications

(8 citation statements)

References 20 publications

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“…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].…”

Section: Characteristic Decompositionmentioning

confidence: 99%

“…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.…”

Section: And the Conclusion Follows □mentioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Comprehensive Characteristic Decomposition of Parametric Polynomial Systems

Dong

Mou

et al. 2021

Proceedings of the 2021 International Symposium on Symbolic and Algebraic Computation

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This paper presents an algorithm that decomposes an arbitrary set F of multivariate polynomials involving parameters into finitely many sets Γ i of (lexicographical) Gröbner bases G i j such that associated with each Γ i there is a system A i of parametric constraints, all the A i 's partition the parameter space, all the G i j 's in each Γ i remain Gröbner bases under specialization of the parameters satisfying the constraints in A i , and for each i the Gröbner bases G i j together with their corresponding W-characteristic sets form a normal characteristic decomposition of F . The sets of Gröbner bases computed by the algorithm provide a comprehensive characteristic decomposition of F that is structure-invariant under the associated constraints and possesses many algebraic and geometric properties, including most of the notable properties on comprehensive Gröbner systems and comprehensive triangular decomposition. Some of these properties are highlighted in the paper and advantages of the proposed algorithm are discussed briefly from the aspects of methodological simplicity and computational performance using illustrative examples and preliminary experiments. CCS CONCEPTS• Computing methodologies → Symbolic and algebraic algorithms; Algebraic algorithms.

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Section: Characteristic Decompositionmentioning

confidence: 99%

Section: And the Conclusion Follows □mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Comprehensive Characteristic Decomposition of Parametric Polynomial Systems

Dong

Mou

et al. 2021

Proceedings of the 2021 International Symposium on Symbolic and Algebraic Computation

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“…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. 下面分别介绍这两种分解算法.…”

Section: 特征分解算法unclassified

“…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].…”

Section: R)unclassified