Comprehensive Triangular Decomposition

Chen, Changbo; Golubitsky, Oleg; Lemaire, François; Maza, Marc Moreno; Pan, Wei

doi:10.1007/978-3-540-75187-8_7

Cited by 71 publications

(101 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…The equivalence between the other points is a consequence of [8 Comparison of Theorem 1 with other works. Inspecting the proof of Lemma 6, we see that Property 3 is equivalent to [19, Definition 1.3 (normal ascending chains)], which refers to [20], i.e.…”

Section: On the Regularity Property Of Polynomialsmentioning

confidence: 58%

“…However, the proof of that Theorem just refers to [8] and [18] and thus misses the use of [5,Theorem 1.6].…”

Section: On the Regularity Property Of Polynomialsmentioning

confidence: 99%

“…For computational purposes, it is desirable to avoid computing the resultant with respect to x k of polynomials which do not both depend on x k , as in [8,Definition 4] and [15, Definition 1]. The iterated resultant is then defined as follows:…”

Section: On the Regularity Property Of Polynomialsmentioning

confidence: 99%

See 2 more Smart Citations

On the Regularity Property of Differential Polynomials Modulo Regular Differential Chains

Boulier

Lemaire

Sedoglavic

2011

Computer Algebra in Scientific Computing

Self Cite

View full text Add to dashboard Cite

Section: On the Regularity Property Of Polynomialsmentioning

confidence: 58%

“…However, the proof of that Theorem just refers to [8] and [18] and thus misses the use of [5,Theorem 1.6].…”

Section: On the Regularity Property Of Polynomialsmentioning

confidence: 99%

See 1 more Smart Citation

On the Regularity Property of Differential Polynomials Modulo Regular Differential Chains

Boulier

Lemaire

Sedoglavic

2011

Computer Algebra in Scientific Computing

Self Cite

View full text Add to dashboard Cite

“…The concept of comprehensive triangular decomposition was introduced in [5] in order to study the specialization properties of regular chains. As a byproduct, one can determine for which parameter values, a parametric polynomial system has complex solutions, or, in general, compute the projection of a constructible set, or do quantifier elimination over an algebraically closed field.…”

Section: Introductionmentioning

confidence: 99%

On Solving Parametric Polynomial Systems

2012

View full text Add to dashboard Cite

Abstract. In the authors' previous work, the concept of comprehensive triangular decomposition of parametric semi-algebraic systems (RCTD for short) was introduced. For a given parametric semi-algebraic system, say S, an RCTD partitions the parametric space into disjoint semialgebraic sets, above each of which the real solutions of S are described by a finite family of triangular systems. Such a decomposition permits to easily count the number of distinct real solutions depending on different parameter values as well as to conveniently describe the real solutions as continuous functions of the parameters. In this paper, we present the implementation of RCTD in the RegularChains library, namely the RealComprehensiveTriangularize command. The use of RCTD is illustrated by the stability analysis of several biological systems.

show abstract

“…The input constructible sets of RefiningPartition are assumed to be represented by regular systems. For any other forms, one should use triangular decompositions [3] to obtain such a representation. The work in [7] suggests that the techniques from multiple-valued logic minimization could help with simplifying the problems before triangular decomposition.…”

Section: Introductionmentioning

confidence: 99%