Thomas Decomposition of Algebraic and Differential Systems

Bächler, Thomas; Gerdt, Vladimir P.; Lange-Hegermann, Markus; Robertz, Daniel

doi:10.1007/978-3-642-15274-0_4

Cited by 16 publications

(30 citation statements)

References 25 publications

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“…, n. When involving no parameter, system (4.1) can be effectively solved by means of Gröbner bases [7] or triangular decomposition [27,51] and hence the solution number may be obtained. For solving parametric systems, there are several methods such as 1. the traversal method, which traverses all the possible parameters and solves the resulting parameter-free systems, 2. the method of triangular sets for Boolean systems [20], 3. the methods of comprehensive Gröbner bases and comprehensive Gröbner systems [31,40,47], 4. the methods of characteristic sets with projection [19], regular systems [43], simple systems [3,29], and comprehensive triangular decomposition [9].…”

Section: Detection Of Steady States For Finite Dynamical Systemsmentioning

confidence: 99%

“…We have tried quantifier elimination for the above-mentioned six quantified formulas using the available software package QEPCAD, 3 functions Resolve and Reduce in Mathematica, routines in Mathematica as described in [38], and package REDLOG 4 in REDUCE (which are implemented on the basis of the methods of CAD [10,11] and virtual term substitution [48,49]). The numbers of steady states and stable steady states could be computed in Mathematica in tens of seconds.…”

Section: Illustrative Examplementioning

confidence: 99%

See 1 more Smart Citation

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: Detection Of Steady States For Finite Dynamical Systemsmentioning

confidence: 99%

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 algorithm has been implemented in Maple [BLH10]. Efficiency strongly depends on certain choices and optimizations in the implementation, such as tuning the order in which polynomials are considered and limiting coefficient growth by systematic reduction of coefficients.…”

Section: Methodsmentioning

confidence: 99%

Thomas decomposition and its applications

Bächler

Gerdt

Lange-Hegermann

et al. 2011

ACM Commun. Comput. Algebra

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Nowadays, triangular decomposition algorithms and software implementing them have become powerful tools for symbolic treatment of systems of multivariate polynomial and (non-linear) partial differential equations. In our ISSAC 2010 poster, we present the algebraic and differential Thomas decomposition and applications of our new Maple implementation [BLH10]. Algebraic Thomas DecompositionAmong the various triangular decompositions the Thomas one stands by itself. It was suggested by the American mathematician J.M.Thomas [Tho37, Tho62] and decomposes a finite system of polynomial equations and inequations into finitely many triangular subsystems that he called simple. The work was continued by Wang [Wan98, Wan01] and Gerdt [Ger08]. Unlike other decomposition algorithms it yields a disjoint decomposition of the zeroes. Every simple system is a regular chain, it defines a characterizable radical ideal and reduction decides membership of polynomials in this ideal. CountingThe disjointness of the Thomas decomposition and the properties of simple systems allow constructing a polynomial -the counting polynomial -which counts the (possibly infinite) set of solutions of a polynomial system via iterated fibrations of projections [Ple09]. Unlike the Hilbert polynomial, the counting polynomial disregards multiplicities of solutions. Differential Thomas DecompositionThe differential Thomas decomposition is concerned with manipulations of polynomial differential equations and inequations. It makes systems simple by treating them as algebraic systems for the differential variables. Moreover, the systems are completed to involution in the sense of Janet [Jan29] to find all integrability conditions. As above, the disjointness allows counting of solutions. Furthermore, a suitable ranking enables the solving of differential elimination problems. Algorithmic AspectsUnlike other triangular decompositions, the Thomas decomposition requires disjointness of solutions and several restrictions regarding the inequations. This forces us to treat inequations as an

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“…This differential-algebraic equation can now be simplified, by means of a differential elimination software, such as the DifferentialAlgebra package or the recent [1]. With the DifferentialAlgebra package, the output involves three cases.…”

Section: Approximating Modelsmentioning

confidence: 99%