In engineering and applied mathematics, polynomial systems arise whose solution sets contain components of different dimensions and multiplicities. In this article we present algorithms, based on homotopy continuation, that compute much of the geometric information contained in the primary decomposition of the solution set. In particular, ignoring multiplicities, our algorithms lay out the decomposition of the set of solutions into irreducible components, by finding, at each dimension, generic points on each component. As by-products, the computation also determines the degree of each component and an upper bound on itsmultiplicity. The bound issharp (i.e., equal to one) for reduced components. The algorithms make essential use of generic projection and interpolation, and can, if desired, describe each irreducible component precisely as the common zeroesof a finite number of polynomials.

Many applications modeled by polynomial systems have positive dimensional solution components (e.g., the path synthesis problems for four-bar mechanisms) that are challenging to compute numerically by homotopy continuation methods. A procedure of A. Sommese and C. Wampler consists in slicing the components with linear subspaces in general position to obtain generic points of the components as the isolated solutions of an auxiliary system. Since this requires the solution of a number of larger overdetermined systems, the procedure is computationally expensive and also wasteful because many solution paths diverge. In this article an embedding of the original polynomial system is presented, which leads to a sequence of homotopies, with solution paths leading to generic points of all components as the isolated solutions of an auxiliary system. The new procedure significantly reduces the number of paths to solutions that need to be followed. This approach has been implemented and applied to...

One of the most important applications of Buchberger's algorithm for Gröbner basis computation [BUC1,2,4] is the solution of systems of polynomial equations (having finitely many roots)...

PHoM is a software package in C++ for finding all isolated solutions of polynomial systems using a polyhedral homotopy continuation method. Among three modules constituting the package, the first module StartSystem constructs a family of polyhedral-linear homotopy functions, based on the polyhedral homotopy theory, from input data for a given system of polynomial equations f (x) = 0. The second module CMPSc traces the solution curves of the homotopy equations to compute all isolated solutions of f (x) = 0. The third module Verify checks whether all isolated solutions of f (x) = 0 have been approximated correctly. We describe numerical methods used in each module and the usage of the package. Numerical results to demonstrate the performance of PHoM include some large polynomial systems that have not been solved previously.

In this paper we describe some experimental findings on selection strategies for Gröbner basis computation with the Buchberger algorithm. In particular, the results suggest that the "sugar flavor" of the "normal selection", implemented first in CoCoA, then in AlPI, and now in SCRATCHPAD-II, is the best choice for a selection strategy. It has to be combined with the "straight-forward" simplification strategy and with a special form of the Gebauer-Möller criteria to obtain the best results. The idea of the "sugar flavor" is the following: the Buchberger algorithm for homogeneous ideals, with degree-compatible term ordering and normal selection strategy, usually works fine. Homogenizing the basis of the ideal is good for the strategy, but bad for the basis to be computed. The sugar flavor computes, for every polynomial in the course of the algorithm, "the degree that it would have if computed with the homogeneous algorithm", and uses this phantom degree (the sugar) only for the selection strategy. We have tested several examples with different selection strategies, and the sugar flavor has proved to be always the best choice or very near to it. The comparison between the different variants of the sugar flavor has been made, but the results are up to now inconclusive. We include a complete deterministic description of the Buchberger algorithm as it was used in our experiments.

Abstract not found

Monomial representations and operations for Grobner bases computations are investigated from an implementation point of view. The technique of vectorized monomial operations is introduced and it is shown how it expedites computations of Grobner bases. Furthermore, a rank-based monomial representation and comparison technique is examined and it is concluded that this technique does not yield an additional speedup over vectorized comparisons. Extensive benchmark tests with the Computer Algebra System Singular are used to evaluate these concepts. 1 Introduction The method of Grobner bases (GB) (see, for example, [8] for an introduction) is undoubtly one of the most important and prominent success stories of the field of Computer Algebra. Starting in the 1960's, an unsolved problem has developed into an essential computational tool with a great variety of applications and more and more powerful implementations. The heart of the GB method are computations of Grobner or Standard bases with...

this paper we describe two parallel formulations of Buchberger algorithm, one for y

The SymbolicData project has the following three main goals: 1. to systematically collect existing symbolic computation benchmark data and to produce tools to extend and maintain this collection; 2. to design and implement concepts for trusted benchmarks computations on the collected data; and 3. to provide tools for data access/selection/transformation using di erent technologies. SymbolicData has developed from a \grass root initiative" of a small number of people to a stage where it should be presented to, and evaluated and used by a wider community. In this paper we report about the current state of the project, i.e., we describe the main design principles and tools which were developed to realize our goals. 1

Abstract not found