Parallel Homotopy Algorithms to Solve Polynomial Systems

2006 ◽  
pp. 225-234 ◽  
Author(s):  
Anton Leykin ◽  
Jan Verschelde ◽  
Yan Zhuang
Keyword(s):  
Polynomial Systems ◽  
Homotopy Algorithms

scholarly journals Generalized Linear Product Homotopy Algorithms and the Computation of Reachable Surfaces

2004 ◽  
Vol 4 (3) ◽  
pp. 226-234 ◽  
Author(s):  
Hai-Jun Su ◽  
J. Michael McCarthy ◽  
Layne T. Watson
Keyword(s):  
Polynomial Systems ◽  
The Body ◽  
Algebraic Surfaces ◽  
Degree Of Freedom ◽  
Homotopy Algorithm ◽  
Spatial Configurations ◽  
Spherical Wrist ◽  
Homotopy Algorithms ◽  
Three Degree Of Freedom ◽  
Serial Chains

In this paper, we apply a homotopy algorithm to the problem of finding points in a moving body that lie on specific algebraic surfaces for a given set of spatial configurations of the body. This problem is a generalization of Burmester’s determination of points in a body that lie on a circle for five planar positions. We focus on seven surfaces that we term “reachable” because they correspond to serial chains with two degree-of-freedom positioning structures combined with a three degree-of-freedom spherical wrist. A homotopy algorithm based on generalized linear products is used to provide a convenient estimate of the number of solutions of these polynomial systems. A parallelized version of this algorithm was then used to numerically determine all of the solutions.

The Convex Polytopes and Homogeneous Coordinate Rings of Bivariate Polynomials

2019 ◽  
Vol 16 (2) ◽  
pp. 1
Author(s):  
Shamsatun Nahar Ahmad ◽  
Nor’Aini Aris ◽  
Azlina Jumadi
Keyword(s):  
Convex Polytopes ◽  
Polynomial Systems ◽  
Matrix Formulation ◽  
Bivariate Polynomials ◽  
Homogeneous Coordinates ◽  
The Matrix ◽  
Projective Vector ◽  
Coordinate Rings ◽  
Resultant Matrix ◽  
The Given

Concepts from algebraic geometry such as cones and fans are related to toric varieties and can be applied to determine the convex polytopes and homogeneous coordinate rings of multivariate polynomial systems. The homogeneous coordinates of a system in its projective vector space can be associated with the entries of the resultant matrix of the system under consideration. This paper presents some conditions for the homogeneous coordinates of a certain system of bivariate polynomials through the construction and implementation of the Sylvester-Bèzout hybrid resultant matrix formulation. This basis of the implementation of the Bèzout block applies a combinatorial approach on a set of linear inequalities, named 5-rule. The inequalities involved the set of exponent vectors of the monomials of the system and the entries of the matrix are determined from the coefficients of facets variable known as brackets. The approach can determine the homogeneous coordinates of the given system and the entries of the Bèzout block. Conditions for determining the homogeneous coordinates are also given and proven.

scholarly journals On the first fall degree of summation polynomials

2019 ◽  
Vol 13 (3-4) ◽  
pp. 229-237
Author(s):  
Stavros Kousidis ◽  
Andreas Wiemers
Keyword(s):  
Elliptic Curve ◽  
Gröbner Basis ◽  
Discrete Logarithm ◽  
Polynomial Systems ◽  
Discrete Logarithm Problem ◽  
Grobner Basis ◽  
Weil Descent ◽  
Degree Bound

Abstract We improve on the first fall degree bound of polynomial systems that arise from a Weil descent along Semaev’s summation polynomials relevant to the solution of the Elliptic Curve Discrete Logarithm Problem via Gröbner basis algorithms.

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