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Triangular decompositions of polynomial systems

Marc Moreno Maza
2006 Proceedings of the 2006 international symposium on Symbolic and algebraic computation - ISSAC '06  
How to pass from one triangular decomposition to another?  ...  They form a triangular decomposition of V (F ). 29 From triangular to equiprojectable decomposition NOTATION. Let V (F ) ⊂ A n (K) be finite with F ⊂ K[x 1 , . . . , x n ].  ...  . • Unfortunately, most polynomial systems F ⊆ Q[X] (both in theory and practice) are equiprojectable, that is they can be represented by a single regular chain. • However, for F ⊆ Z/pZ [X] where p prime  ... 
doi:10.1145/1145768.1145776 dblp:conf/issac/Maza06 fatcat:fds2pw66qrblrjgy5hr6x562ia

Algorithms for Computing Triangular Decompositions of Polynomial Systems [article]

Changbo Chen, Marc Moreno Maza
2011 arXiv   pre-print
We propose new algorithms for computing triangular decompositions of polynomial systems incrementally.  ...  With respect to previous works, our improvements are based on a weakened notion of a polynomial GCD modulo a regular chain, which permits to greatly simplify and optimize the sub-algorithms.  ...  The authors would like to thank the support of Maplesoft, Mitacs and Nserc of Canada.  ... 
arXiv:1104.0689v1 fatcat:ez5kkazvrrgpdjpstb5iosqorm

Algorithms for computing triangular decomposition of polynomial systems

Changbo Chen, Marc Moreno Maza
2012 Journal of symbolic computation  
We then present new techniques that we regard as essential to the recent success and for future research directions in the development of triangular decomposition methods.  ...  We discuss algorithmic advances which have extended the pioneer work of Wu on triangular decompositions.  ...  Algorithms for computing triangular decompositions of polynomial systems can be classified in several ways.  ... 
doi:10.1016/j.jsc.2011.12.023 fatcat:gc2zppoymre67bsldpoduufc6e

Algorithms for computing triangular decompositions of polynomial systems

Changbo Chen, Marc Moreno Maza
2011 Proceedings of the 36th international symposium on Symbolic and algebraic computation - ISSAC '11  
We then present new techniques that we regard as essential to the recent success and for future research directions in the development of triangular decomposition methods.  ...  We discuss algorithmic advances which have extended the pioneer work of Wu on triangular decompositions.  ...  Algorithms for computing triangular decompositions of polynomial systems can be classified in several ways.  ... 
doi:10.1145/1993886.1993904 dblp:conf/issac/ChenM11 fatcat:l4pki44enrawnhrvnxomxns4r4

Multiplicity-preserving triangular set decomposition of two polynomials

Jin-San Cheng, Xiao-Shan Gao
2014 Journal of Systems Science and Complexity  
In this paper, a multiplicity-preserving triangular set decomposition algorithm is proposed for a system of two polynomials, which involves only computing the primitive polynomial remainder sequence of  ...  In the bivariate case, the authors give a complete algorithm to decompose the system into zeros represented by triangular sets with multiplicities.  ...  And there are some other methods to compute multiplicities of zeros of polynomial systems, which are not by triangular set theories, for example, [25] .  ... 
doi:10.1007/s11424-014-2017-0 fatcat:22ueextldjfylgs4ftxdx7sriu

A triangular decomposition algorithm for differential polynomial systems with elementary computation complexity

Wei Zhu, Xiao-Shan Gao
2016 Journal of Systems Science and Complexity  
The key idea is to eliminate one algebraic variable from a set of polynomials in one step using the theory of multivariate resultant.  ...  In this paper, a new triangular decomposition algorithm is proposed for ordinary differential polynomial systems, which has triple exponential computational complexity.  ...  . , Ψ t /M ) is one component of the output, then according to the algorithm it comes from a procedure like (9) and (10).  ... 
doi:10.1007/s11424-016-5040-5 fatcat:twombmuz7jb3dcgvdjlzolwjpq

Fast arithmetic for triangular sets: From theory to practice

Xin Li, Marc Moreno Maza, Éric Schost
2009 Journal of symbolic computation  
We study arithmetic operations for triangular families of polynomials, concentrating on multiplication in dimension zero.  ...  By a suitable extension of fast univariate Euclidean division, we obtain theoretical and practical improvements over a direct recursive approach; for a family of special cases, we reach quasi-linear complexity  ...  This article is an expanded version of (20) .  ... 
doi:10.1016/j.jsc.2008.04.019 fatcat:qz2mqpvfxrdytodh4ujr7pz2mq

Solving Parametric Polynomial Systems by RealComprehensiveTriangularize [chapter]

Changbo Chen, Marc Moreno Maza
2014 Lecture Notes in Computer Science  
of triangular systems.  ...  In the authors' previous work, the concept of comprehensive triangular decomposition of parametric semi-algebraic systems (RCTD for short) was introduced.  ...  Related notions and commands The notion of a regular chain, introduced independently in [10] and [16] , is closely related to that of a triangular decomposition of a polynomial system.  ... 
doi:10.1007/978-3-662-44199-2_76 fatcat:ygbg6k6zrjholnrjerbk36px44

ORDERING IN SOLVING SYSTEMS OF EQUATIONS

KUN JIANG, XIAO-SHAN GAO
2000 Computer Mathematics  
In this paper, we present an algorithm of O(n 3 ) to find such a "good" order using algorithms from graph theory.  ...  To use a "good" variable order is one of the effective ways to prevent the occurrence of large polynomials in an elimination method.  ...  Algorithm with Mixed Order Wu-Ritt's zero decomposition algorithm is used to decompose the zero set of a set of polynomial equations into the union of zero sets of polynomial equations in triangular form  ... 
doi:10.1142/9789812791962_0042 fatcat:fqjjoz6fzzaw3pu2ymbiicik3e

Decomposing Polynomial Systems into Simple Systems

D. Wang
1998 Journal of symbolic computation  
This paper presents a method that decomposes any pair of polynomial sets into finitely many simple systems with an associated zero decomposition.  ...  A simple system is a pair of multivariate polynomial sets (one set for equations and the other for inequations) ordered in triangular form, in which every polynomial is squarefree and has non-vanishing  ...  Acknowledgements Part of this paper was written in February/March 1994 when the author was visiting the MM Research Center of Academia Sinica, under the support of the K. C.  ... 
doi:10.1006/jsco.1997.0177 fatcat:l7zzbktt2nhwfjrbmptmaguosy

Generic Regular Decompositions for Generic Zero-Dimensional Systems [article]

Xiaoxian Tang, Zhenghong Chen, Bican Xia
2013 arXiv   pre-print
The algorithm has been implemented with Maple 15 and experimented with a number of benchmarks from the literature. Empirical results are also presented to show the good performance of the algorithm.  ...  The so called weakly relatively simplicial decomposition plays a crucial role in the algorithm, which is based on the theories of subresultant chains.  ...  We would like to thank Changbo Chen and Yao Sun for providing a great deal of test-systems. Thanks also go to Rong Xiao for his suggestions.  ... 
arXiv:1208.6112v4 fatcat:uvea26lh6zeqxorwa7qf5j7eje

Real Solving for Positive Dimensional Systems

Philippe Aubry, Fabrice Rouillier, Mohab Safey El Din
2002 Journal of symbolic computation  
By studying the critical points of the restriction to the variety of the distance function to one well chosen point, we show how to provide a set of zerodimensional systems whose zeros contain at least  ...  From the output of our algorithm, one can then apply, for each computed zerodimensional system, any symbolic or numerical algorithm for counting or approximating the real solutions.  ...  Acknowledgements We would like to thank J.-C. Faugère, D. Lazard and M.-F. Roy for their helpful comments, advice and support and H.  ... 
doi:10.1006/jsco.2002.0563 fatcat:w4ja4f2o5bcedfwvwnguqnwvle

Generic regular decompositions for parametric polynomial systems

Zhenghong Chen, Xiaoxian Tang, Bican Xia
2015 Journal of Systems Science and Complexity  
In this paper, the two concepts, generic regular decomposition (GRD) and regular-decomposition-unstable (RDU) variety introduced in [19] for generic zero-dimensional systems, are extended to the case where  ...  An algorithm is provided to compute GRDs and the associated RDU varieties of parametric systems simultaneously on the basis of the algorithm for generic zero-dimensional systems proposed in [19] .  ...  Thank Changbo Chen and Yao Sun for providing a great deal of test-systems.  ... 
doi:10.1007/s11424-015-3015-6 fatcat:p2vb4yls6vg7hmltroyboons6a

Generic Regular Decompositions for Parametric Polynomial Systems [article]

Zhenghong Chen, Xiaoxian Tang, Bican Xia
2013 arXiv   pre-print
In this paper, the two concepts, generic regular decomposition (GRD) and regular-decomposition-unstable (RDU) variety introduced in [19] for generic zero-dimensional systems, are extended to the case where  ...  An algorithm is provided to compute GRDs and the associated RDU varieties of parametric systems simultaneously on the basis of the algorithm for generic zero-dimensional systems proposed in [19].  ...  Thank Changbo Chen and Yao Sun for providing a great deal of test-systems.  ... 
arXiv:1301.3991v1 fatcat:uvbfty3cgzdk3m6ekbxhhcyb34

Fast algorithms, modular methods, parallel approaches and software engineering for solving polynomial systems symbolically

Yuzhen Xie
2007 ACM Communications in Computer Algebra  
We first adapt nearly optimal algorithms for polynomial arithmetic over fields to direct products of fields for polynomial multiplication, inversion and GCD computations.  ...  In this thesis, we study and apply fast algorithms, modular methods, parallel approaches and software engineering techniques to improve the efficiency of symbolic solvers for computing triangular decomposition  ...  However, this leads to computations which, in practice and in theory, are much more expensive than those of triangular decompositions.  ... 
doi:10.1145/1358190.1358195 fatcat:qck42ui4rzcgti62umwt2njmje
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