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38
On the Theories of Triangular Sets
 J. SYMB. COMP
, 1999
"... Different notions of triangular sets are presented. The relationship between these notions are studied. The main result is that four different existing notions of good triangular sets are equivalent. ..."
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Cited by 85 (35 self)
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Different notions of triangular sets are presented. The relationship between these notions are studied. The main result is that four different existing notions of good triangular sets are equivalent.
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 ..."
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Cited by 69 (37 self)
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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.
Well known theorems on triangular systems and the D5 principle
"... The theorems that we present in this paper are very important to prove the correctness of triangular decomposition algorithms. The most important of them are not new but their proofs are. We illustrate how they articulate with the D5 principle. ..."
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Cited by 32 (18 self)
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The theorems that we present in this paper are very important to prove the correctness of triangular decomposition algorithms. The most important of them are not new but their proofs are. We illustrate how they articulate with the D5 principle.
Using Galois Ideals For Computing Relative Resolvents
 J. Symb. Comp
, 1998
"... . In this paper we establish that some ideals which occur in Galois theory are generated by a triangular set of polynomials. This geometric property seems important for the development of symbolic methods in Galois theory. It may be exploited in order to obtain more efficient algorithms. Actually, i ..."
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Cited by 30 (4 self)
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. In this paper we establish that some ideals which occur in Galois theory are generated by a triangular set of polynomials. This geometric property seems important for the development of symbolic methods in Galois theory. It may be exploited in order to obtain more efficient algorithms. Actually, it enables us to present here a new algebraic method for computing relative resolvents which works with any kind of invariant. 1. Introduction Let k be a perfect field and ¯ k an algebraic closure of k. Let f be a univariate polynomial of k[X] supposed separable with degree n, and\Omega be an ordered set of the n roots of f in ¯ k n . In [25] is introduced the notion of ideal of\Omega\Gamma32615/39 invariant by a subset L of the symmetric group of degree n. It generalizes the notion of ideal of relations and the notion of ideal of symmetric relations. We call them Galois ideals. This paper presents two important results. First, we prove in Theorem 5.5 that a Galois ideal associated w...
Decomposition plans for geometric constraint systems
 J. Symbolic Computation
, 2001
"... A central issue in dealing with geometric constraint systems for CAD/CAM/CAE is the generation of an optimal decomposition plan that not only aids efficient solution, but also captures design intent and supports conceptual design. Though complex, this issue has evolved and crystallized over the past ..."
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Cited by 24 (0 self)
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A central issue in dealing with geometric constraint systems for CAD/CAM/CAE is the generation of an optimal decomposition plan that not only aids efficient solution, but also captures design intent and supports conceptual design. Though complex, this issue has evolved and crystallized over the past few years, permitting us to take the next important step: in this paper, we formalize, motivate and explain the decomposition–recombination (DR)planning problem as well as several performance measures by which DRplanning algorithms can be analyzed and compared. These measures include: generality, validity, completeness, Church–Rosser property, complexity, best and worstchoice approximation factors, (strict) solvability preservation, ability to deal with underconstrained systems, and ability to incorporate conceptual design decompositions specified by the designer. The problem and several of the performance measures are formally defined here for the first time—they closely reflect specific requirements of CAD/CAM applications. The clear formulation of the problem and performance measures allow us to precisely analyze and compare existing DRplanners that use two wellknown types of decomposition methods: SR (constraint shape recognition) and MM (generalized maximum matching) on constraint graphs. This analysis additionally serves to illustrate and provide intuitive substance to the newly formalized measures. In Part II of this article, we use the new performance measures to guide the development of a new DRplanning algorithm which excels with respect to these performance measures. c ○ 2001 Academic Press 1.
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 ..."
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Cited by 17 (11 self)
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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.
Change of ordering for regular chains in positive dimension
 IN ILIAS S. KOTSIREAS, EDITOR, MAPLE CONFERENCE 2006
, 2006
"... We discuss changing the variable ordering for a regular chain in positive dimension. This quite general question has applications going from implicitization problems to the symbolic resolution of some systems of differential algebraic equations. We propose a modular method, reducing the problem to d ..."
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Cited by 16 (7 self)
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We discuss changing the variable ordering for a regular chain in positive dimension. This quite general question has applications going from implicitization problems to the symbolic resolution of some systems of differential algebraic equations. We propose a modular method, reducing the problem to dimension zero and using NewtonHensel lifting techniques. The problems raised by the choice of the specialization points, the lack of the (crucial) information of what are the free and algebraic variables for the new ordering, and the efficiency regarding the other methods are discussed. Strong hypotheses (but not unusual) for the initial regular chain are required. Change of ordering in dimension zero is taken as a subroutine.
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 ..."
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Cited by 16 (14 self)
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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.
Real Solving for Positive Dimensional Systems
 IN "JOURNAL OF SYMBOLIC COMPUTATION
, 2000
"... Finding one point on each semialgebraically connected component of a real algebraic variety, or at least deciding if such a variety is empty or not, is a fundamental problem of computational real algebraic geometry. Even though numerous studies have been done on the subject, only a few number of ..."
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Cited by 13 (2 self)
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Finding one point on each semialgebraically connected component of a real algebraic variety, or at least deciding if such a variety is empty or not, is a fundamental problem of computational real algebraic geometry. Even though numerous studies have been done on the subject, only a few number of efficient implementations exists. In this paper, we propose a new efficient and practical algorithm for computing such points. 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 zeroes contain at least one point on each semialgebraically connected component of the studied variety, without any assumption neither on the variety (smoothness or compactness for example) nor on the system of equations that define it. Once such a result is computed, one can then apply, for each computed zerodimensional system, any symbolic or numerical algorithm for counting or approx...
On computerassisted classification of coupled integrable equations
 J. Symb. Comp
, 2002
"... equations ..."