On Characteristic Decomposition and Quasi-characteristic Decomposition

Dong, Rina; Mou, Chenqi

doi:10.1007/978-3-030-26831-2_9

Cited by 3 publications

(4 citation statements)

References 28 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…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%

“…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 1 more Smart Citation

Comprehensive Characteristic Decomposition of Parametric Polynomial Systems

Dong

Mou

et al. 2021

Proceedings of the 2021 International Symposium on Symbolic and Algebraic Computation

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Section: Characteristic Decompositionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Comprehensive Characteristic Decomposition of Parametric Polynomial Systems

Dong

Mou

et al. 2021

Proceedings of the 2021 International Symposium on Symbolic and Algebraic Computation

Self Cite

View full text Add to dashboard Cite

show abstract

“…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.…”

Section: Introductionmentioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

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

Li¹,

Xia²

2022

Preprint

View full text Add to dashboard Cite

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 Gröbner 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 triangulardecomposition 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. CCS CONCEPTS• Computing methodologies → Algebraic algorithms.

show abstract

Squarefree normal representation of zeros of zero-dimensional polynomial systems

Xu,

Wang,

2024

Journal of Symbolic Computation

View full text Add to dashboard Cite

On Characteristic Decomposition and Quasi-characteristic Decomposition

Cited by 3 publications

References 28 publications

Comprehensive Characteristic Decomposition of Parametric Polynomial Systems

Comprehensive Characteristic Decomposition of Parametric Polynomial Systems

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

Squarefree normal representation of zeros of zero-dimensional polynomial systems

Contact Info

Product

Resources

About