scholarly journals Approximate Gröbner Bases, Overdetermined Polynomial Systems, and Approximate GCDs

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
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.

Solving Polynomial Systems for the Kinematic Analysis and Synthesis of Mechanisms and Robot Manipulators

1995 ◽  
Vol 117 (B) ◽  
pp. 71-79 ◽  
Author(s):  
M. Raghavan ◽  
B. Roth
Keyword(s):  
Kinematic Analysis ◽  
State Of The Art ◽  
Polynomial Systems ◽  
The State ◽  
Polynomial Equations ◽  
Systems Of Equations ◽  
Synthesis Of Mechanisms ◽  
Analysis And Synthesis ◽  
Systems Of Polynomial Equations ◽  
And Robotics

Problems in mechanisms analysis and synthesis and robotics lead naturally to systems of polynomial equations. This paper reviews the state of the art in the solution of such systems of equations. Three well-known methods for solving systems of polynomial equations, viz., Dialytic Elimination, Polynomial Continuation, and Grobner bases are reviewed. The methods are illustrated by means of simple examples. We also review important kinematic analysis and synthesis problems and their solutions using these mathematical procedures.

scholarly journals A Signature-Based Algorithm for Computing Gröbner Bases over Principal Ideal Domains

2019 ◽  
Vol 14 (2) ◽  
pp. 515-530
Author(s):  
Maria Francis ◽  
Thibaut Verron
Keyword(s):  
Principal Ideal ◽  
Gröbner Bases ◽  
General Setting ◽  
Polynomial Systems ◽  
Regular Sequence ◽  
Grobner Bases ◽  
Integral Domains ◽  
Unique Factorization Domain ◽  
Univariate Polynomials ◽  
Principal Ideal Domains

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).

Solving polynomial equations for chemical problems using Gröbner bases

2004 ◽  
Vol 102 (23-24) ◽  
pp. 2521-2535 ◽  
Author(s):  
Manfred Minimair * ◽  
Michael P. Barnett †
Keyword(s):  
Gröbner Bases ◽  
Grobner Bases ◽  
Polynomial Equations

scholarly journals An extension of Buchberger’s criteria for Gröbner basis decision

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.

Numerical Continuation Methods for Solving Polynomial Systems Arising in Kinematics

1990 ◽  
Vol 112 (1) ◽  
pp. 59-68 ◽  
Author(s):  
C. W. Wampler ◽  
A. P. Morgan ◽  
A. J. Sommese
Keyword(s):  
Polynomial Systems ◽  
Numerical Continuation ◽  
Continuation Methods ◽  
Basic Material ◽  
Polynomial Equations ◽  
All Solutions ◽  
Recent Developments ◽  
Position Problem ◽  
Main Ideas ◽  
Systems Of Polynomial Equations

Many problems in mechanism design and theoretical kinematics can be formulated as systems of polynomial equations. Recent developments in numerical continuation have led to algorithms that compute all solutions to polynomial systems of moderate size. Despite the immediate relevance of these methods, they are unfamiliar to most kinematicians. This paper attempts to bridge that gap by presenting a tutorial on the main ideas of polynomial continuation along with a section surveying advanced techniques. A seven position Burmester problem serves to illustrate the basic material and the inverse position problem for general six-axis manipulators shows the usefulness of the advanced techniques.

Gröbner Bases for Polynomial Equations

2006 ◽  
pp. 922-924
Author(s):  
P. O. Lindberg ◽  
Lars Svensson
Keyword(s):  
Gröbner Bases ◽  
Grobner Bases ◽  
Polynomial Equations
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