Triangular decomposition of semi-algebraic systems

Chen, Changbo; Davenport, James H.; May, John P.; Maza, Marc Moreno; Xia, Bican; Xiao, Rong

doi:10.1016/j.jsc.2011.12.014

Cited by 58 publications

(52 citation statements)

References 36 publications

(27 reference statements)

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“…The two semi-algebraic systems above may be solved by using the method of Yang and Xia [53] for real solution classification (implemented as a Maple package DISCOVERER 1 by Xia, see also recent improvements in [8]), or the method of discriminant varieties of Lazard and Rouillier [28] (implemented as a Maple package DV 2 by Moroz and Rouillier). The obtained results of classification for the numbers of (stable) steady states are shown in Table 2.…”

Section: Illustrative Examplementioning

confidence: 99%

Stability Analysis for Discrete Biological Models Using Algebraic Methods

Mou

Wei

et al. 2011

Math.Comput.Sci.

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This paper is concerned with stability analysis of biological networks modeled as discrete and finite dynamical systems. We show how to use algebraic methods based on quantifier elimination, real solution classification and discriminant varieties to detect steady states and to analyze their stability and bifurcations for discrete dynamical systems. For finite dynamical systems, methods based on Gröbner bases and triangular sets are applied to detect steady states. The feasibility of our approach is demonstrated by the analysis of stability and bifurcations of several discrete biological models using implementations of algebraic methods.

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Section: Illustrative Examplementioning

confidence: 99%

Stability Analysis for Discrete Biological Models Using Algebraic Methods

Mou

Wei

et al. 2011

Math.Comput.Sci.

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“…The double exponential cost of this algorithm [8], is a major barrier to its application. See [6] and [5] for modern improvements using triangular decompositions.…”

Section: Real Algebraic Geometrymentioning

confidence: 99%

Finding points on real solution components and applications to differential polynomial systems

Reid

2013

Proceedings of the 38th International Symposium on Symbolic and Algebraic Computation

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In this paper we extend complex homotopy methods to finding witness points on the irreducible components of real varieties. In particular we construct such witness points as the isolated real solutions of a constrained optimization problem.First a random hyperplane characterized by its random normal vector is chosen. Witness points are computed by a polyhedral homotopy method. Some of them are at the intersection of this hyperplane with the components. Other witness points are the local critical points of the distance from the plane to components. A method is also given for constructing regular witness points on components, when the critical points are singular.The method is applicable to systems satisfying certain regularity conditions. Illustrative examples are given. We show that the method can be used in the consistent initialization phase of a popular method due to Pryce and Pantelides for preprocessing differential algebraic equations for numerical solution.

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“…This algorithm is based on properties of isolated points of real algebraic sets and computation of real radicals of zero-dimensional ideals. Instead of computing real radicals, Chen et al (2010Chen et al ( , 2013Chen et al ( , 2011 give a method to decompose semi-algebraic systems into regular semi-algebraic systems.…”

Section: Introductionmentioning

confidence: 99%

On the Complexity of Computing Real Radicals of Polynomial Systems

Din

Yang

Zhi

2018

Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation

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Let f = (f 1 ,. .. , f s) be a sequence of polynomials in Q[X 1 ,. .. , X n ] of maximal degree D and V ⊂ C n be the algebraic set defined by f and r be its dimension. The real radical re f associated to f is the largest ideal which defines the real trace of V. When V is smooth, we show that re f , has a finite set of generators with degrees bounded by deg V. Moreover, we present a probabilistic algorithm of complexity (snD n) O(1) to compute the minimal primes of re f. When V is not smooth, we give a probabilistic algorithm of complexity s O(1) (nD) O(nr2 r) to compute rational parametrizations for all irreducible components of the real algebraic set V ∩ R n. Experiments are given to show the efficiency of our approaches.

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Triangular decomposition of semi-algebraic systems

Cited by 58 publications

References 36 publications

Stability Analysis for Discrete Biological Models Using Algebraic Methods

Stability Analysis for Discrete Biological Models Using Algebraic Methods

Finding points on real solution components and applications to differential polynomial systems

On the Complexity of Computing Real Radicals of Polynomial Systems

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