Cylindrical algebraic decomposition is one of the most important tools for computing with semi-algebraic 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 F-invariant cylindrical decomposition of the n-dimensional complex space, from which we extract an F-invariant cylindrical algebraic decomposition of the n-dimensional real space. We report on an implementation of this new approach for constructing cylindrical algebraic decompositions.

Regular chains and triangular decompositions are fundamental and well-developed tools for describing the complex solutions of polynomial systems. This paper proposes adaptations of these tools focusing on solutions of the real analogue: semi-algebraic systems. We show that any such system can be decomposed into finitely many regular semi-algebraic systems. We propose two specifications of such a decomposition and present corresponding algorithms. Under some assumptions, one type of decomposition can be computed in singly exponential time w.r.t. the number of variables. We implement our algorithms and the experimental results illustrate their effectiveness.

Given a regular chain T, we aim at finding an efficient way for computing a system of generators of sat(T), the saturated ideal of T. A natural idea is to test whether the equality 〈T 〉 = sat(T) holds, that is, whether T generates its saturated ideal. By generalizing the notion of primitivity from univariate polynomials to regular chains, we establish a necessary and sufficient condition, together with a Gröbner basis free algorithm, for testing this equality. Our experimental results illustrate the efficiency of this approach in practice.

Abstract. We study continuous dynamical systems defined by autonomous ordinary differential equations, themselves given by parametric rational functions. For such systems, we provide semi-algebraic descriptions of their hyperbolic and non-hyperbolic equilibria, their asymptotically stable hyperbolic equilibria, their Hopf bifurcations. To this end, we revisit various criteria on sign conditions for the roots of a real parametric univariate polynomial. In addition, we introduce the notion of comprehensive triangular decomposition of a semi-algebraic system and demonstrate that it is well adapted for our study. 1

It is well known that multivariate nonlinear polynomial systems are genuinely harder to solve than linear systems. This is intrinsic, as the solutions display features that do not manifest themselves in the linear case. For instance, the solution set of a nonlinear polynomial system may consist of components of different dimensions. Moreover, even if all components have the same dimension, say dimension zero, they may not be glued into a single component without loosing finer properties such as equi-projectability, which is important in the design of algorithms for solving polynomial systems. To illustrate this latter property without defining it formally, consider the following system of three equations, two unknowns and with 750,000 complex solutions: {x 1000 − 1, (x 500 − 1)(y 500 − 1), y 1000 − 1} (1) The algorithm presented in [9] and implemented by the Triangularize command of the RegularChains library produces the following triangular decomposition

Symbolic methods are powerful tools in scientific computing. The implementation of symbolic solvers is, however, a highly difficult task due to the extremely high time and space complexity of the problem. 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, one of the most promising methods for solving non-linear systems of equations symbolically. We first adapt nearly optimal algorithms for polynomial arithmetic over fields to direct products of fields for polynomial multiplication, inversion and GCD compu-tations. Then, by introducing the notion of equiprojectable decomposition, a sharp modular method for triangular decompositions based on Hensel lifting techniques is obtained. Its implementation also brings to the Maple computer algebra system a unique capacity for automatic case discussion and recombination. A high-level categorical parallel framework is developed, written in the Al-dor language, to support high-performance computer algebra on symmetric multi-

(with redundancy) (without redundancy and the D ′ j s refine the C ′ i s)

�Solving systems of polynomial equations with parameters symbolically is in demand for an increasing number of applications such as dynamic systems and optimization. �ParametricSystemTools is a new module of the Regular-

This article is a continuation of our earlier work [3], which introduced triangular decompositions of semi-algebraic systems and algorithms for computing them. Our new contributions include theoretical results based on which we obtain practical improvements for these decomposition algorithms. We exhibit new results on the theory of border polynomials of parametric semi-algebraic systems: in particular a geometric characterization of its“true boundary”(Definition 2). In order to optimize these algorithms, we also propose a technique, that we call relaxation, which can simplify the decomposition process and reduce the number of redundant components in the output. Moreover, we present procedures for basic set-theoretical operations on semi-algebraic sets represented by triangular decomposition. Experimentation confirms the effectiveness of our techniques. Categories andSubject Descriptors

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