scholarly journals SHARPER COMPLEXITY BOUNDS FOR ZERO-DIMENSIONAL GRÖBNER BASES AND POLYNOMIAL SYSTEM SOLVING

2011 ◽  
Vol 21 (05) ◽  
pp. 703-713 ◽  
Author(s):  
AMIR HASHEMI ◽  
DANIEL LAZARD
Keyword(s):  
Polynomial System ◽  
Gröbner Bases ◽  
Mean Value ◽  
Arithmetic Mean ◽  
Grobner Bases ◽  
Maximal Degree ◽  
Complexity Bounds ◽  
Polynomial System Solving ◽  
Polynomial Ideals ◽  
Dense Representation

The main purpose of this paper is to improve the bound of complexity of the well-known algorithms on polynomial ideals having complexities polynomial in dn, where d is the maximal degree of input polynomials and n is the number of variables. Instead of this bound, we present the more accurate bound max {S, Dn} where S is the size of the input polynomials in dense representation, and D is the arithmetic mean value of the degrees of input polynomials.

On generating sets and gröbner bases for polynomial ideals

1992 ◽  
Vol 20 (8) ◽  
pp. 2271-2287 ◽  
Author(s):  
Henrik Bresinsky ◽  
Frank Curtis
Keyword(s):  
Gröbner Bases ◽  
Grobner Bases ◽  
Generating Sets ◽  
Polynomial Ideals

Gröbner Bases for Polynomial Ideals

1992 ◽  
pp. 429-471
Author(s):  
K. O. Geddes ◽  
S. R. Czapor ◽  
G. Labahn
Keyword(s):  
Gröbner Bases ◽  
Grobner Bases ◽  
Polynomial Ideals

scholarly journals Gröbner bases and primary decomposition of polynomial ideals

1988 ◽  
Vol 6 (2-3) ◽  
pp. 149-167 ◽  
Author(s):  
Patrizia Gianni ◽  
Barry Trager ◽  
Gail Zacharias
Keyword(s):  
Gröbner Bases ◽  
Grobner Bases ◽  
Primary Decomposition ◽  
Polynomial Ideals

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

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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