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.

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.

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.

. 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...

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 DR-planning algorithms can be analyzed and compared. These measures include: generality, validity, completeness, Church–Rosser property, complexity, best- and worst-choice 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 DR-planners that use two well-known 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 DR-planning algorithm which excels with respect to these performance measures. c ○ 2001 Academic Press 1.

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 Newton-Hensel 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.

The talk gives a survey on some symbolic algorithmic methods for solving systems of algebraic equations with special emphasis on parametric systems. Besides complex solutions I consider also real solutions of systems including inequalities. The techniques described include the Euclidean algorithm, Gröbner bases, characteristic sets, univariate and multivariate Sturm-Sylvester theorems, comprehensive Grobner bases and elimination methods for parametric optimization problems. Some examples illustrate the use of symbolic algorithms for the solution of parametric systems.

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 pre-existing Maple implementations, as well as more optimized Magma functions. In most cases, our new code outperforms the other packages by several orders of magnitude.

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Finding one point on each semi-algebraically 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 zero-dimensional systems whose zeroes contain at least one point on each semi-algebraically 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 zero-dimensional system, any symbolic or numerical algorithm for counting or approx...