Real Solution Classification for Parametric Semi-Algebraic Systems
نویسندگان
چکیده
Real solution classification of parametric polynomial systems is a crucial problem in real quantifier elimination which interests Volker Weispfenning and others. In this paper, we present a stepwise refinement algorithm for real solution classifications of a class of parametric systems consisting of polynomial equations, inequalities and inequations. For an input system, the algorithm outputs the necessary and sufficient conditions (in terms of quantifier-free formulae) on the parameters for the system to have a given number of real solutions. Although the algorithm makes use of a pcad algorithm, it is different from any existing methods.
منابع مشابه
Computing Equilibria of Semi-algebraic Economies Using Triangular Decomposition and Real Solution Classification
In this paper, we are concerned with the problem of determining the existence of multiple equilibria in economic models. We propose a general and complete approach for identifying multiplicities of equilibria in semi-algebraic economies, which may be expressed as semi-algebraic systems. The approach is based on triangular decomposition and real solution classification, two powerful tools of alg...
متن کاملComputing with semi-algebraic sets: Relaxation techniques and effective boundaries
We discuss parametric polynomial systems, with algorithms for real root classification and triangular decomposition of semi-algebraic systems as our main applications. We exhibit new results in the theory of border polynomials of parametric semi-algebraic systems: in particular a geometric characterization of its “true boundary” (Definition 1). In order to optimize the corresponding decompositi...
متن کاملSolving Parametric Polynomial Systems by RealComprehensiveTriangularize
In the authors’ previous work, the concept of comprehensive triangular decomposition of parametric semi-algebraic systems (RCTD for short) was introduced. For a given parametric semi-algebraic system, say S, an RCTD partitions the parametric space into disjoint semialgebraic sets, above each of which the real solutions of S are described by a finite family of triangular systems. Such a decompos...
متن کاملA Semi-Algebraic Approach for the Computation of Lyapunov Functions
In this paper we deal with the problem of computing Lyapunov functions for stability verification of differential systems. We concern on symbolic methods and start the discussion with a classical quantifier elimination model for computing Lyapunov functions in a given polynomial form, especially in quadratic forms. Then we propose a new semi-algebraic method by making advantage of the local pro...
متن کاملSemi-algebraic Description of the Equilibria of Dynamical Systems
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 r...
متن کامل