D. A. Yanovich

Construction of Janet Bases I. Monomial Bases

Algorithms for computation of Janet bases for monomial ideals and implementation of these algorithms are presented. As data structures for finite monomial sets the binary trees called Janet trees are selected. An algorithm for construction of a Janet basis for the ideal generated by a finite monomial set is described. This algorithm contains as subalgorithms those to search for Janet divisor in a given tree and to insert monomials into the tree in the process of completion to involution. The algorithms presented have been implemented in C in the form of package for completion of monomial sets to Janet involutive ones. An example is given to illustrate practical efficiency of the monomial algorithms and their implementation.

Fast Search for the Janet Divisor

An algorithm for fast search for the involutive monomial Janet divisor is suggested. Such search is an important part of the construction of monomial and polynomial Janet bases. For a data structure for a finite set of monomials, the binary tree is taken, which reflects properties of the Janet division.

Effectiveness of involutive criteria in computation of polynomial Janet bases

In this paper, effectiveness of involutive criteria in the elimination of useless prolongations when computing polynomial Janet bases, which are typical representatives of involutive bases, is discussed. One of the results of this study is that the role of the criteria in an involutive algorithm is not as important as in the Buchberger algorithm. It is shown also that these criteria affect the growth of intermediate coefficients.

Parallel computation of Janet and Gröbner bases over rational numbers

In this paper, a parallel algorithm for computation of polynomial Janet bases is considered. After construction of a Janet basis, a reduced Gröbner basis (which is a subset of the Janet basis) is extracted from it without any additional reductions. The algorithm discussed is an improved version of an earlier suggested parallel algorithm. The efficiency of a C implementation of the algorithm and its scalability are illustrated by way of the standard test examples that are often used for comparing various algorithms and codes for computing Gröbner bases.

Parallelization of an Algorithm for Computation of Involutive Janet Bases

A parallelization of an involutive algorithm for computation of Janet bases for polynomial ideals is considered. The algorithm is shown to admit an efficient parallelization with the use of shared-memory architecture. Results of parallelization of modular computations on a two-processor computer are presented.

Efficiency estimate for distributed computation of Gröbner bases and involutive bases

Several years ago, we presented a program complex for parallel computation of Gröbner bases that works on computers with shared-memory architecture. Unfortunately, the number of the processors that we can use is small (from 2 to 16) because of hardware constraints. This paper presents a program for distributed computation of bases that relies on the same principles but works in a network consisting of heterogeneous machines. The effectiveness of such an approach is estimated from the standpoint of the processor capacity usage and the required network bandwidth, and methods optimizing usage of these resources are specified.

Parallel modular computation of Gröbner and involutive bases

An overview of an algorithm and an efficient implementation of parallel computing of involutive and Gröbner bases with the help of modular operations is presented. Difficulties arising in modulo calculations and in the reconstruction of a basis with coefficients in ℤ by its modular images are considered; Some ways to overcome these difficulties are indicated.

Implementation of the FGLM Algorithm and Finding Roots of Polynomial Involutive Systems

In this paper, an implementation of the FGLM algorithm that transforms Gröbner bases from one ordering to another is presented. Some additional optimizations that considerably expedite computations are considered. It is shown that this algorithm can be used for finding roots of polynomial systems represented in the involutive form.

Compact representation of polynomials for algorithms for computing Gröbner and involutive bases

In the computation of involutive and Gröbner bases with rational coefficients, the major part of the memory is occupied by precision numbers; however, in the case of modular operations (especially, in the computation of Gröbner bases), of most importance is the problem of compact representation of monomials composing polynomials of the system. For this purpose, for example, ZDD diagrams and other structures are used, which make execution of typical operations—multiplication by a monomial and reduction of polynomials—more complicated. In this paper, an attempt is made to develop convenient (in the sense of computation of bases) and compact representation of polynomials that is based on hash-tables. Results of test runs are presented.