Triangular decomposition of semi-algebraic systems

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

doi:10.1145/1837934.1837972

Cited by 33 publications

(26 citation statements)

References 25 publications

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“…If the coefficients of S are real numbers and if only the real solutions are required (in which case S is said to be semi-algebraic), then those real solutions can be obtained by a triangular decomposition into so-called regular semi-algebraic systems, a notion introduced in [4]. In both cases, each of these simpler systems has a triangular shape and remarkable properties, which justifies the terminology.…”

Section: Related Notions and Commandsmentioning

confidence: 93%

“…When the purpose is to describe all the solutions of S, whether their coordinates are real numbers or not, (in which case S is said to be algebraic) those simpler systems are required to be regular chains 4 . If the coefficients of S are real numbers and if only the real solutions are required (in which case S is said to be semi-algebraic), then those real solutions can be obtained by a triangular decomposition into so-called regular semi-algebraic systems, a notion introduced in [4].…”

Section: Related Notions and Commandsmentioning

confidence: 99%

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On Solving Parametric Polynomial Systems

2012

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Abstract. 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 decomposition permits to easily count the number of distinct real solutions depending on different parameter values as well as to conveniently describe the real solutions as continuous functions of the parameters. In this paper, we present the implementation of RCTD in the RegularChains library, namely the RealComprehensiveTriangularize command. The use of RCTD is illustrated by the stability analysis of several biological systems.

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Section: Related Notions and Commandsmentioning

confidence: 93%

Section: Related Notions and Commandsmentioning

confidence: 99%

On Solving Parametric Polynomial Systems

2012

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“…Recent work by Chen et al (2013) proposes adaptations of these tools to the real analogue: semi-algebraic systems. They describe two algorithms to decompose any real polynomial system into finitely many regular semialgebraic systems.…”

Section: Parametric Analysis For K 19mentioning

confidence: 99%

Identifying the parametric occurrence of multiple steady states for some biological networks

Bradford

Davenport

England

et al. 2020

Journal of Symbolic Computation

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We consider a problem from biological network analysis of determining regions in a parameter space over which there are multiple steady states for positive real values of variables and parameters. We describe multiple approaches to address the problem using tools from Symbolic Computation. We describe how progress was made to achieve semi-algebraic descriptions of the multistationarity regions of parameter space, and compare symbolic results to numerical methods.The biological networks studied are models of the mitogen-activated protein kinases (MAPK) network which has already consumed considerable effort using special insights into its structure of corresponding models. Our main example is a model with 11 equations in 11 variables and 19 parameters, 3 of which are of interest for symbolic treatment. The model also imposes positivity conditions on all variables and parameters.We apply combinations of symbolic computation methods designed for mixed equality/inequality systems, specifically virtual substitution, lazy real triangularization and cylindrical algebraic decomposition, as well as a simplification technique adapted from Gaussian elimination and graph theory.We are able to determine multistationarity of our main example over a 2-dimensional parameter space. We also study a second MAPK model and a symbolic grid sampling technique which can locate such regions in 3-dimensional parameter space.

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“…One of the difficulties we face with using triangular sets is that most of the prior work on them has been with C[z], the polynomials with complex coefficients. Our concern is only with the real solutions of a system of polynomials in Q[z] which has seen recent development [3,4]. We'll develop the necessary machinery for working with Nash manifolds but first we'll look at a standard solution method over C.…”

Section: Definition 42 (Triangular Sets)mentioning

confidence: 99%

Whitney’s Theorem, Triangular Sets, and Probabilistic Descent on Manifolds

Dreisigmeyer

2019

J Optim Theory Appl

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We examine doing probabilistic descent over manifolds implicitly defined by a set of polynomials with rational coefficients. The system of polynomials is assumed to be triangularized. An application of Whitney's embedding theorem allows us to work in a reduced dimensional embedding space. A numerical continuation method applied to the reduced-dimensional embedded manifold is used to drive the procedure.

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Triangular decomposition of semi-algebraic systems

Cited by 33 publications

References 25 publications

On Solving Parametric Polynomial Systems

On Solving Parametric Polynomial Systems

Identifying the parametric occurrence of multiple steady states for some biological networks

Whitney’s Theorem, Triangular Sets, and Probabilistic Descent on Manifolds

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