Results 1  10
of
34
On triangular decompositions of algebraic varieties
 Presented at the MEGA2000 Conference
, 1999
"... We propose an efficient algorithm for computing triangular decompositions of algebraic varieties. It is based on an incremental process and produces components in order of decreasing dimension. The combination of these two major features is obtained by means of lazy evaluation techniques and a lifti ..."
Abstract

Cited by 81 (35 self)
 Add to MetaCart
(Show Context)
We propose an efficient algorithm for computing triangular decompositions of algebraic varieties. It is based on an incremental process and produces components in order of decreasing dimension. The combination of these two major features is obtained by means of lazy evaluation techniques and a lifting property for calculations modulo regular chains. This allows a good management of the intermediate computations, as confirmed by several implementations and applications of this work. Our algorithm is also well suited for parallel execution.
Computing Cylindrical Algebraic Decomposition via Triangular Decomposition
, 2009
"... Cylindrical algebraic decomposition is one of the most important tools for computing with semialgebraic sets, while triangular decomposition is among the most important approaches for manipulating constructible sets. In this paper, for an arbitrary finite set F ⊂ R[y1,..., yn] we apply comprehensiv ..."
Abstract

Cited by 51 (18 self)
 Add to MetaCart
(Show Context)
Cylindrical algebraic decomposition is one of the most important tools for computing with semialgebraic sets, while triangular decomposition is among the most important approaches for manipulating constructible sets. In this paper, for an arbitrary finite set F ⊂ R[y1,..., yn] we apply comprehensive triangular decomposition in order to obtain an Finvariant cylindrical decomposition of the ndimensional complex space, from which we extract an Finvariant cylindrical algebraic decomposition of the ndimensional real space. We report on an implementation of this new approach for constructing cylindrical algebraic decompositions.
On the complexity of the D5 principle
 In Proc. of Transgressive Computing 2006
, 2006
"... The D5 Principle was introduced in 1985 by Jean Della Dora, Claire Dicrescenzo and Dominique Duval in their celebrated note “About a new method for computing in algebraic number fields”. This innovative approach automatizes reasoning based on case discussion and is also known as “Dynamic Evaluation” ..."
Abstract

Cited by 33 (19 self)
 Add to MetaCart
The D5 Principle was introduced in 1985 by Jean Della Dora, Claire Dicrescenzo and Dominique Duval in their celebrated note “About a new method for computing in algebraic number fields”. This innovative approach automatizes reasoning based on case discussion and is also known as “Dynamic Evaluation”. Applications of the D5 Principle have been made in Algebra, Computer Algebra, Geometry and Logic. Many algorithms for solving polynomial systems symbolically need to perform standard operations, such as GCD computations, over coefficient rings that are direct products of fields rather than fields. We show in this paper how asymptotically fast algorithms for polynomials over fields can be adapted to this more general context, thanks to the D5 Principle. 1
Algorithms for Computing Triangular Decomposition of Polynomial Systems
, 2011
"... We discuss algorithmic advances which have extended the pioneer work of Wu on triangular decompositions. We start with an overview of the key ideas which have led to either better implementation techniques or a better understanding of the underlying theory. We then present new techniques that we reg ..."
Abstract

Cited by 29 (21 self)
 Add to MetaCart
We discuss algorithmic advances which have extended the pioneer work of Wu on triangular decompositions. We start with an overview of the key ideas which have led to either better implementation techniques or a better understanding of the underlying theory. 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.
Computation of canonical forms for ternary cubics
 in ISSAC. 2000
, 2002
"... In this paper we conduct a careful study of the equivalence classes of ternary cubics under general complex linear changes of variables. Our new results are based on the method of moving frames and involve triangular decompositions of algebraic varieties. We provide a computationally efficient algor ..."
Abstract

Cited by 21 (8 self)
 Add to MetaCart
(Show Context)
In this paper we conduct a careful study of the equivalence classes of ternary cubics under general complex linear changes of variables. Our new results are based on the method of moving frames and involve triangular decompositions of algebraic varieties. We provide a computationally efficient algorithm that matches an arbitrary ternary cubic with its canonical form and explicitly computes a corresponding linear change of coordinates. We also describe a classification of the symmetry groups of ternary cubics.
Computations modulo regular chains
 In Proc. of ISSAC’09
, 2009
"... The computation of triangular decompositions involves two fundamental operations: polynomial GCDs modulo regular chains and regularity test modulo saturated ideals. We propose new algorithms for these core operations based on modular methods and fast polynomial arithmetic. We rely on new results con ..."
Abstract

Cited by 19 (10 self)
 Add to MetaCart
(Show Context)
The computation of triangular decompositions involves two fundamental operations: polynomial GCDs modulo regular chains and regularity test modulo saturated ideals. We propose new algorithms for these core operations based on modular methods and fast polynomial arithmetic. We rely on new results connecting polynomial subresultants and GCDs modulo regular chains. We report on extensive experimentation, comparing our code to preexisting Maple implementations, as well as more optimized Magma functions. In most cases, our new code outperforms the other packages by several orders of magnitude.
On computerassisted classification of coupled integrable equations
 J. Symb. Comp
, 2002
"... equations ..."
(Show Context)
Verification of polynomial system solvers
 In Proceedings of AWFS 2007
, 2007
"... Abstract. We discuss the verification of mathematical software solving polynomial systems symbolically by way of triangular decomposition. Standard verification techniques are highly resource consuming and apply only to polynomial systems which are easy to solve. We exhibit a new approach which mani ..."
Abstract

Cited by 9 (8 self)
 Add to MetaCart
(Show Context)
Abstract. We discuss the verification of mathematical software solving polynomial systems symbolically by way of triangular decomposition. Standard verification techniques are highly resource consuming and apply only to polynomial systems which are easy to solve. We exhibit a new approach which manipulates constructible sets represented by regular systems. We provide comparative benchmarks of different verification procedures applied to four solvers on a large set of wellknown polynomial systems. Our experimental results illustrate the high efficiency of our new approach. In particular, we are able to verify triangular decompositions of polynomial systems which are not easy to solve. Key words: Software verification, polynomial system solver, triangular decomposition. 1