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22
Computing the Limit Points of the Quasicomponent of a Regular Chain in Dimension One
"... For a regular chain R in dimension one, we propose an algorithm which computes the (nontrivial) limit points of the quasicomponent of R, that is, the set W (R) \ W (R). Our procedure relies on Puiseux series expansions and does not require to compute a system of generators of the saturated ideal ..."
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Cited by 5 (2 self)
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For a regular chain R in dimension one, we propose an algorithm which computes the (nontrivial) limit points of the quasicomponent of R, that is, the set W (R) \ W (R). Our procedure relies on Puiseux series expansions and does not require to compute a system of generators of the saturated ideal of R. We provide experimental results illustrating the benefits of our algorithms.
Solving polynomial systems via triangular decomposition
, 2011
"... Finding the solutions of a polynomial system is a fundamental problem with numerous applications in both the academic and industrial world. In this thesis, we target on computing symbolically both the real and the complex solutions of nonlinear polynomial systems with or without parameters. To this ..."
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Cited by 4 (2 self)
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Finding the solutions of a polynomial system is a fundamental problem with numerous applications in both the academic and industrial world. In this thesis, we target on computing symbolically both the real and the complex solutions of nonlinear polynomial systems with or without parameters. To this end, we improve existing algorithms for computing triangular decompositions. Based on that, we develop various new tools for solving polynomial systems and illustrate their effectiveness by applications. We propose new algorithms for computing 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 subalgorithms. Extracting common work from similar expensive computations is also a key feature of our algorithms. We adapt the concepts of regular chain and triangular decomposition, originally designed for studying the complex solutions of polynomial systems, to describing the
Relative equilibria in the fourvortex problem with two pairs of equal vorticities
 Journal of Nonlinear Science
"... Abstract We examine in detail the relative equilibria in the planar fourvortex problem where two pairs of vortices have equal strength, that is, Γ 1 = Γ 2 = 1 and Γ 3 = Γ 4 = m where m ∈ R −{0} is a parameter. One main result is that, for m > 0, the convex configurations all contain a line of s ..."
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Abstract We examine in detail the relative equilibria in the planar fourvortex problem where two pairs of vortices have equal strength, that is, Γ 1 = Γ 2 = 1 and Γ 3 = Γ 4 = m where m ∈ R −{0} is a parameter. One main result is that, for m > 0, the convex configurations all contain a line of symmetry, forming a rhombus or an isosceles trapezoid. The rhombus solutions exist for all m but the isosceles trapezoid case exists only when m is positive. In fact, there exist asymmetric convex configurations when m < 0. In contrast to the Newtonian fourbody problem with two equal pairs of masses, where the symmetry of all convex central configurations is unproven, the equations in the vortex case are easier to handle, allowing for a complete classification of all solutions. Precise counts on the number and type of solutions (equivalence classes) for different values of m, as well as a description of some of the bifurcations that occur, are provided. Our techniques involve a combination of analysis, and modern and computational algebraic geometry. Communicated by P. Newton.
LAYERED CYLINDRICAL ALGEBRAIC DECOMPOSITION
"... Abstract. In this paper the idea of a Layered CAD is introduced, a truncation of a CAD to only high dimensional cells. Limiting to fulldimensional cells has already been investigated in the literature, but including another level can be beneficial for applications. A related topological property is ..."
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Cited by 3 (3 self)
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Abstract. In this paper the idea of a Layered CAD is introduced, a truncation of a CAD to only high dimensional cells. Limiting to fulldimensional cells has already been investigated in the literature, but including another level can be beneficial for applications. A related topological property is defined and related to robotics motion planning. The distribution of cell dimensions in a CAD is investigated and layered CAD ideas are combined with other research.
Semialgebraic description of the equilibria of dynamical systems
"... Abstract. We study continuous dynamical systems defined by autonomous ordinary differential equations, themselves given by parametric rational functions. For such systems, we provide semialgebraic descriptions of their hyperbolic and nonhyperbolic equilibria, their asymptotically stable hyperbolic ..."
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Abstract. We study continuous dynamical systems defined by autonomous ordinary differential equations, themselves given by parametric rational functions. For such systems, we provide semialgebraic descriptions of their hyperbolic and nonhyperbolic 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 semialgebraic system and demonstrate that it is well adapted for our study. 1
On solving parametric polynomial systems
"... triangular decomposition, regular chain. Abstract. Border polynomial and discriminant variety are two important notions related to parametric polynomial system solving, in particular, for partitioning the parameter values into regions where the solutions of the system depend continuously on the para ..."
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triangular decomposition, regular chain. Abstract. Border polynomial and discriminant variety are two important notions related to parametric polynomial system solving, in particular, for partitioning the parameter values into regions where the solutions of the system depend continuously on the parameters. In this paper, we study the relations between those notions in the case of parametric triangular systems. We also investigate the properties and computation of the nonproperness locus of the canonical projection restricted at a parametric regular chain or at its saturated ideal.
Computing with SemiAlgebraic Sets: Relaxation Techniques and Effective Boundaries
"... We discuss parametric polynomial systems, with algorithms for real root classification and triangular decomposition of semialgebraic systems as our main applications. We exhibit new results in the theory of border polynomials of parametric semialgebraic systems: in particular a geometric character ..."
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We discuss parametric polynomial systems, with algorithms for real root classification and triangular decomposition of semialgebraic systems as our main applications. We exhibit new results in the theory of border polynomials of parametric semialgebraic systems: in particular a geometric characterization of its “true boundary ” (Definition 1). In order to optimize the corresponding decomposition algorithms, we also propose a technique, that we call relaxation, which can simplify the decomposition process and reduce the number of components in the output. This paper extends our earlier works [6, 7]. Key words: triangular decomposition, regular semialgebraic system, border polynomial, effective boundary, relaxation. 1.
An Algorithm for Computing the Limit Points of the Quasicomponent of a Regular Chain
, 2013
"... For a regular chain R, we propose an algorithm which computes the (nontrivial) limit points of the quasicomponent of R, that is, the set W (R) \W (R). Our procedure relies on Puiseux series expansions and does not require to compute a system of generators of the saturated ideal of R. We focus on t ..."
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For a regular chain R, we propose an algorithm which computes the (nontrivial) limit points of the quasicomponent of R, that is, the set W (R) \W (R). Our procedure relies on Puiseux series expansions and does not require to compute a system of generators of the saturated ideal of R. We focus on the case where this saturated ideal has dimension one and we discuss extensions of this work in higher dimensions. We provide experimental results illustrating the benefits of our algorithms.