Article

Comprehensive triangular decomposition

Authors:

Oleg Golubitsky,

François Lemaire,

Marc Moreno Maza,

Wei PanAuthors Info & Claims

CASC'07: Proceedings of the 10th international conference on Computer Algebra in Scientific Computing

Pages 73 - 101

Published: 16 September 2007 Publication History

Abstract

We introduce the concept of comprehensive triangular decomposition (CTD) for a parametric polynomial system F with coefficients in a field. In broad words, this is a finite partition of the the parameter space into regions, so that within each region the "geometry" (number of irreducible components together with their dimensions and degrees) of the algebraic variety of the specialized system F(u) is the same for all values u of the parameters.

We propose an algorithm for computing the CTD of F. It relies on a procedure for solving the following set theoretical instance of the coprime factorization problem. Given a family of constructible sets A₁,..., A_s, compute a family B₁,..., B_t of pairwise disjoint constructible sets, such that for all 1 ≤ i ≤ s the set A_i writes as a union of some of the B₁,...,B_t.

We report on an implementation of our algorithm computing CTDs, based on the RegularChains library in MAPLE. We provide comparative benchmarks with MAPLE implementations of related methods for solving parametric polynomial systems. Our results illustrate the good performances of our CTD code.

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Cited By

Chen CMoreno Maza M(2016)Quantifier elimination by cylindrical algebraic decomposition based on regular chainsJournal of Symbolic Computation10.1016/j.jsc.2015.11.00875:C(74-93)Online publication date: 1-Jul-2016
https://dl.acm.org/doi/10.1016/j.jsc.2015.11.008
Guo FSafey El Din MWang CZhi LRobertz DLinton SYokoyama K(2015)Optimizing a Parametric Linear Function over a Non-compact Real Algebraic VarietyProceedings of the 2015 ACM International Symposium on Symbolic and Algebraic Computation10.1145/2755996.2756666(205-212)Online publication date: 24-Jun-2015
https://dl.acm.org/doi/10.1145/2755996.2756666
Alvandi PChen CHashemi AMaza M(2015)Regular Chains under Linear Changes of Coordinates and ApplicationsProceedings of the 17th International Workshop on Computer Algebra in Scientific Computing - Volume 930110.1007/978-3-319-24021-3_3(30-44)Online publication date: 14-Sep-2015
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Comprehensive triangular decomposition
1. Theory of computation

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Published In

cover image Guide Proceedings

CASC'07: Proceedings of the 10th international conference on Computer Algebra in Scientific Computing

September 2007

459 pages

ISBN:3540751866

Editors:
Victor G. Ganzha
Technische Universität München, Institut für Informatik, Garching, Germany
,
Ernst W. Mayr
Technische Universität München, Institut für Informatik, Garching, Germany
,
Evgenii V. Vorozhtsov
Russian Academy of Sciences, Institute of Theoretical and Applied Mechanics, Novosibirsk, Russia

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Springer-Verlag

Berlin, Heidelberg

Publication History

Published: 16 September 2007

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Cited By

Chen CMoreno Maza M(2016)Quantifier elimination by cylindrical algebraic decomposition based on regular chainsJournal of Symbolic Computation10.1016/j.jsc.2015.11.00875:C(74-93)Online publication date: 1-Jul-2016
https://dl.acm.org/doi/10.1016/j.jsc.2015.11.008
Guo FSafey El Din MWang CZhi LRobertz DLinton SYokoyama K(2015)Optimizing a Parametric Linear Function over a Non-compact Real Algebraic VarietyProceedings of the 2015 ACM International Symposium on Symbolic and Algebraic Computation10.1145/2755996.2756666(205-212)Online publication date: 24-Jun-2015
https://dl.acm.org/doi/10.1145/2755996.2756666
Alvandi PChen CHashemi AMaza M(2015)Regular Chains under Linear Changes of Coordinates and ApplicationsProceedings of the 17th International Workshop on Computer Algebra in Scientific Computing - Volume 930110.1007/978-3-319-24021-3_3(30-44)Online publication date: 14-Sep-2015
https://dl.acm.org/doi/10.1007/978-3-319-24021-3_3
Chen CMaza MNagasaka KWinkler FSzanto A(2014)Quantifier elimination by cylindrical algebraic decomposition based on regular chainsProceedings of the 39th International Symposium on Symbolic and Algebraic Computation10.1145/2608628.2608666(91-98)Online publication date: 23-Jul-2014
https://dl.acm.org/doi/10.1145/2608628.2608666
Kapur DSun YWang D(2013)An efficient method for computing comprehensive Gröbner basesJournal of Symbolic Computation10.1016/j.jsc.2012.05.01552(124-142)Online publication date: 1-May-2013
https://dl.acm.org/doi/10.1016/j.jsc.2012.05.015
Chen CDavenport JMay JMoreno Maza MXia BXiao R(2013)Triangular decomposition of semi-algebraic systemsJournal of Symbolic Computation10.1016/j.jsc.2011.12.01449(3-26)Online publication date: 1-Feb-2013
https://dl.acm.org/doi/10.1016/j.jsc.2011.12.014
Alvandi PChen CMaza M(2013)Computing the Limit Points of the Quasi-component of a Regular Chain in Dimension OneProceedings of the 15th International Workshop on Computer Algebra in Scientific Computing - Volume 813610.1007/978-3-319-02297-0_3(30-45)Online publication date: 9-Sep-2013
https://dl.acm.org/doi/10.1007/978-3-319-02297-0_3
Chen CMoreno Maza M(2012)Algorithms for computing triangular decomposition of polynomial systemsJournal of Symbolic Computation10.1016/j.jsc.2011.12.02347:6(610-642)Online publication date: 1-Jun-2012
https://dl.acm.org/doi/10.1016/j.jsc.2011.12.023
Beimel AChee YWang HZhang L(2012)Communication-efficient distributed oblivious transferJournal of Computer and System Sciences10.1016/j.jcss.2012.02.00278:4(1142-1157)Online publication date: 1-Jul-2012
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Du XChen ZHu L(2012)Linear complexity of binary sequences derived from Euler quotients with prime-power modulusInformation Processing Letters10.1016/j.ipl.2012.04.011112:14-15(604-609)Online publication date: 1-Aug-2012
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