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Triangular decompositions of polynomial systems

2006
*
Proceedings of the 2006 international symposium on Symbolic and algebraic computation - ISSAC '06
*

How

doi:10.1145/1145768.1145776
dblp:conf/issac/Maza06
fatcat:fds2pw66qrblrjgy5hr6x562ia
*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 ...##
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Algorithms for Computing Triangular Decompositions of Polynomial Systems
[article]

2011
*
arXiv
*
pre-print

We propose new algorithms for computing

arXiv:1104.0689v1
fatcat:ez5kkazvrrgpdjpstb5iosqorm
*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. ...##
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Algorithms for computing triangular decomposition of polynomial systems

2012
*
Journal of symbolic computation
*

We then present new techniques that we regard as essential

doi:10.1016/j.jsc.2011.12.023
fatcat:gc2zppoymre67bsldpoduufc6e
*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. ...##
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Algorithms for computing triangular decompositions of polynomial systems

2011
*
Proceedings of the 36th international symposium on Symbolic and algebraic computation - ISSAC '11
*

We then present new techniques that we regard as essential

doi:10.1145/1993886.1993904
dblp:conf/issac/ChenM11
fatcat:l4pki44enrawnhrvnxomxns4r4
*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. ...##
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Multiplicity-preserving triangular set decomposition of two polynomials

2014
*
Journal of Systems Science and Complexity
*

In this paper, a multiplicity-preserving

doi:10.1007/s11424-014-2017-0
fatcat:22ueextldjfylgs4ftxdx7sriu
*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] . ...##
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A triangular decomposition algorithm for differential polynomial systems with elementary computation complexity

2016
*
Journal of Systems Science and Complexity
*

The key idea is

doi:10.1007/s11424-016-5040-5
fatcat:twombmuz7jb3dcgvdjlzolwjpq
*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). ...##
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Fast arithmetic for triangular sets: From theory to practice

2009
*
Journal of symbolic computation
*

We study arithmetic operations for

doi:10.1016/j.jsc.2008.04.019
fatcat:qz2mqpvfxrdytodh4ujr7pz2mq
*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) . ...##
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Solving Parametric Polynomial Systems by RealComprehensiveTriangularize
[chapter]

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*. ...

##
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ORDERING IN SOLVING SYSTEMS OF EQUATIONS

2000
*
Computer Mathematics
*

In this paper, we present an algorithm

doi:10.1142/9789812791962_0042
fatcat:fqjjoz6fzzaw3pu2ymbiicik3e
*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 ...##
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Decomposing Polynomial Systems into Simple Systems

1998
*
Journal of symbolic computation
*

This paper presents a method that decomposes any pair

doi:10.1006/jsco.1997.0177
fatcat:l7zzbktt2nhwfjrbmptmaguosy
*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. ...##
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Generic Regular Decompositions for Generic Zero-Dimensional Systems
[article]

2013
*
arXiv
*
pre-print

The algorithm has been implemented with Maple 15 and experimented with a number

arXiv:1208.6112v4
fatcat:uvea26lh6zeqxorwa7qf5j7eje
*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. ...##
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Real Solving for Positive Dimensional Systems

2002
*
Journal of symbolic computation
*

By studying the critical points

doi:10.1006/jsco.2002.0563
fatcat:w4ja4f2o5bcedfwvwnguqnwvle
*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. ...##
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Generic regular decompositions for parametric polynomial systems

2015
*
Journal of Systems Science and Complexity
*

In this paper, the two concepts, generic regular

doi:10.1007/s11424-015-3015-6
fatcat:p2vb4yls6vg7hmltroyboons6a
*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*. ...##
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Generic Regular Decompositions for Parametric Polynomial Systems
[article]

2013
*
arXiv
*
pre-print

In this paper, the two concepts, generic regular

arXiv:1301.3991v1
fatcat:uvbfty3cgzdk3m6ekbxhhcyb34
*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*. ...##
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Fast algorithms, modular methods, parallel approaches and software engineering for solving polynomial systems symbolically

2007
*
ACM Communications in Computer Algebra
*

We first adapt nearly optimal algorithms for

doi:10.1145/1358190.1358195
fatcat:qck42ui4rzcgti62umwt2njmje
*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*. ...
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