Gröbner bases for polynomial systems with parameters

AbstractSignature-based algorithms have become a standard approach for Gröbner basis computations for polynomial systems over fields, but how to extend these techniques to coefficients in general rings is not yet as well understood. In this paper, we present a proof-of-concept signature-based algorithm for computing Gröbner bases over commutative integral domains. It is adapted from a general version of Möller’s algorithm (J Symb Comput 6(2–3), 345–359, 1988) which considers reductions by multiple polynomials at each step. This algorithm performs reductions with non-decreasing signatures, and in particular, signature drops do not occur. When the coefficients are from a principal ideal domain (e.g. the ring of integers or the ring of univariate polynomials over a field), we prove correctness and termination of the algorithm, and we show how to use signature properties to implement classic signature-based criteria to eliminate some redundant reductions. In particular, if the input is a regular sequence, the algorithm operates without any reduction to 0. We have written a toy implementation of the algorithm in Magma. Early experimental results suggest that the algorithm might even be correct and terminate in a more general setting, for polynomials over a unique factorization domain (e.g. the ring of multivariate polynomials over a field or a PID).

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An extension of Buchberger’s criteria for Gröbner basis decision

LMS Journal of Computation and Mathematics ◽

10.1112/s1461157008000193 ◽

2010 ◽

Vol 13 ◽

pp. 111-129

Author(s):

John Perry

Keyword(s):

Gröbner Basis ◽

Gröbner Bases ◽

Characterization Theorem ◽

Polynomial Systems ◽

Polynomial Ideal ◽

Grobner Bases ◽

Grobner Basis ◽

Case Scenario ◽

Worst Case ◽

New Criterion

AbstractTwo fundamental questions in the theory of Gröbner bases are decision (‘Is a basisGof a polynomial ideal a Gröbner basis?’) and transformation (‘If it is not, how do we transform it into a Gröbner basis?’) This paper considers the first question. It is well known thatGis a Gröbner basis if and only if a certain set of polynomials (theS-polynomials) satisfy a certain property. In general there arem(m−1)/2 of these, wheremis the number of polynomials inG, but criteria due to Buchberger and others often allow one to consider a smaller number. This paper presents two original results. The first is a new characterization theorem for Gröbner bases that makes use of a new criterion that extends Buchberger’s criteria. The second is the identification of a class of polynomial systemsGfor which the new criterion has dramatic impact, reducing the worst-case scenario fromm(m−1)/2 S-polynomials tom−1.

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Kinematic Analysis of Mechanisms Based on Parametric Polynomial System: Basic Concept of a Method Using Gröbner Cover and Its Application to Planar Mechanisms

Journal of Mechanisms and Robotics ◽

10.1115/1.4042475 ◽

2019 ◽

Vol 11 (2) ◽

Author(s):

Keisuke Arikawa

Keyword(s):

Basic Concept ◽

Polynomial System ◽

Gröbner Bases ◽

Polynomial Systems ◽

Grobner Bases ◽

Truss Structures ◽

Planar Mechanisms ◽

Kinematic Constraints ◽

Parametric Polynomial System ◽

Kinematic Properties

Many kinematic problems in mechanisms can be represented by polynomial systems. By algebraically analyzing the polynomial systems, we can obtain the kinematic properties of the mechanisms. Among these algebraic methods, approaches based on Gröbner bases are effective. Usually, the analyses are performed for specific mechanisms; however, we often encounter phenomena for which, even within the same class of mechanisms, the kinematic properties differ significantly. In this research, we consider the cases where the parameters are included in the polynomial systems. The parameters are used to express link lengths, displacements of active joints, hand positions, and so on. By analyzing a parametric polynomial system (PPS), we intend to comprehensively analyze the kinematic properties of mechanisms represented by these parameters. In the proposed method, we first express the kinematic constraints in the form of PPS. Subsequently, by calculating the Gröbner cover of the PPS, we obtain the segmentation of the parameter space and valid Gröbner bases for each segment. Finally, we interpret the meaning of the segments and their corresponding Gröbner bases. We analyzed planar four- and five-bar linkages and five-bar truss structures using the proposed method. We confirmed that it was possible to enumerate the assembly and working modes and to identify the geometrical conditions that enable overconstrained motions.

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Approximate Gröbner Bases, Overdetermined Polynomial Systems, and Approximate GCDs

ISRN Computational Mathematics ◽

10.1155/2013/352806 ◽

2013 ◽

Vol 2013 ◽

pp. 1-12 ◽

Cited By ~ 2

Author(s):

Daniel Lichtblau

Keyword(s):

Relative Error ◽

Gröbner Bases ◽

Approximate Solutions ◽

Polynomial Systems ◽

Grobner Bases ◽

Polynomial Equations ◽

Secondary Feature ◽

Approximate Polynomial ◽

Polynomial Gcd ◽

Systems Of Polynomial Equations

We discuss computation of Gröbner bases using approximate arithmetic for coefficients. We show how certain considerations of tolerance, corresponding roughly to absolute and relative error from numeric computation, allow us to obtain good approximate solutions to problems that are overdetermined. We provide examples of solving overdetermined systems of polynomial equations. As a secondary feature we show handling of approximate polynomial GCD computations, using benchmarks from the literature.

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Kinematic Analysis of Mechanisms Based on Parametric Polynomial System

Volume 5B: 42nd Mechanisms and Robotics Conference ◽

10.1115/detc2018-85347 ◽

2018 ◽

Author(s):

Keisuke Arikawa

Keyword(s):

Parameter Space ◽

Kinematic Analysis ◽

Polynomial System ◽

Gröbner Bases ◽

Polynomial Systems ◽

Grobner Bases ◽

Algebraic Equations ◽

Kinematic Constraints ◽

Parametric Polynomial System ◽

Planar Linkages

Many kinematic problems of mechanisms can be expressed in the form of polynomial systems. Gröbner Bases computation is effective for algebraically analyzing such systems. In this research, we discuss the cases in which the parameters are included in the polynomial systems. The parameters are used to express the link lengths, the displacements of active joints, hand positions, and so on. By calculating Gröbner Cover of the parametric polynomial system that expresses kinematic constraints, we obtain segmentation of the parameter space and valid Gröbner Bases for each segment. In the application examples, we use planar linkages to interpret the meanings of the algebraic equations that define the segments and the Gröbner Bases. Using these interpretations, we confirmed that it was possible to enumerate the assembly and working modes and to identify the geometrical conditions that enable overconstrained motions.

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