December 5, 1999
In this paper we present an algorithm for construction of minimal involutive polynomial bases which are Groebner bases of the special form. The most general involutive algorithms are based on the concept of involutive monomial division which leads to partition of variables into multiplicative and non-multiplicative. This partition gives thereby the self-consistent computational procedure for constructing an involutive basis by performing non-multiplicative prolongations and multiplicative reductions. Every specific involutive division generates a particular form of involutive computational procedure. In addition to three involutive divisions used by Thomas, Janet and Pommaret for analysis of partial differential equations we define two new ones. These two divisions, as well as Thomas division, do not depend on the order of variables. We prove noetherity, continuity and constructivity of the new divisions that provides correctness and termination of involutive algorithms for any finite set of input polynomials and any admissible monomial ordering. We show that, given an admissible monomial ordering, a monic minimal involutive basis is uniquely defined and thereby can be considered as canonical much like the reduced Groebner basis.
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In this paper we consider an algorithmic technique more general than that proposed by Zharkov and Blinkov for the involutive analysis of polynomial ideals. It is based on a new concept of involutive monomial division which is defined for a monomial set. Such a division provides for each monomial the self-consistent separation of the whole set of variables into two disjoint subsets. They are called multiplicative and non-multiplicative. Given an admissible ordering, this separ...
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In this paper, in addition to the earlier introduced involutive divisions, we consider a new class of divisions induced by admissible monomial orderings. We prove that these divisions are noetherian and constructive. Thereby each of them allows one to compute an involutive Groebner basis of a polynomial ideal by sequentially examining multiplicative reductions of nonmultiplicative prolongations. We study dependence of involutive algorithms on the completion ordering. Based on...
January 8, 2005
In this paper we describe an efficient involutive algorithm for constructing Groebner bases of polynomial ideals. The algorithm is based on the concept of involutive monomial division which restricts the conventional division in a certain way. In the presented algorithm a reduced Groebner basis is the internally fixed subset of an involutive basis, and having computed the later, the former can be output without any extra computational costs. We also discuss some accounts of e...
February 7, 2006
The theory of Groebner Bases originated in the work of Buchberger and is now considered to be one of the most important and useful areas of symbolic computation. A great deal of effort has been put into improving Buchberger's algorithm for computing a Groebner Basis, and indeed in finding alternative methods of computing Groebner Bases. Two of these methods include the Groebner Walk method and the computation of Involutive Bases. By the mid 1980's, Buchberger's work had been ...
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We consider computational and implementation issues for the completion of monomial sets to involution using different involutive divisions. Every of these divisions produces its own completion procedure. For the polynomial case it yields an involutive basis which is a special form of a Groebner basis, generally redundant. We also compare our Mathematica implementation of Janet division to an implementation in C.
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Ihe first author presented an efficient algorithm for computing involutive (and reduced Groebner) bases. In this paper, we consider a modification of this algorithm which simplifies matters to understand it and to implement. We prove correctness and termination of the modified algorithm and also correctness of the used criteria. The proposed algorithm has been implemented in Maple. We present experimental comparison, via some examples, of performance of the modified algorithm...
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In this paper, we describe improved algorithms to compute Janet and Pommaret bases. To this end, based on the method proposed by Moller et al., we present a more efficient variant of Gerdt's algorithm (than the algorithm presented by Gerdt-Hashemi-M.Alizadeh) to compute minimal involutive bases. Further, by using the involutive version of Hilbert driven technique, along with the new variant of Gerdt's algorithm, we modify the algorithm, given by Seiler, to compute a linear ch...
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In this paper we generalize the involutive methods and algorithms devised for polynomial ideals to differential ones generated by a finite set of linear differential polynomials in the differential polynomial ring over a zero characteristic differential field. Given a ranking of derivative terms and an involutive division, we formulate the involutivity conditions which form a basis of involutive algorithms. We present an algorithm for computation of a minimal involutive diffe...
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The new type of ideal basis introduced herein constitutes a compromise between the Gr\"obner bases based on the Buchberger's algorithm and the characteristic sets based on the Wu's method. It reduces the complexity of the traditional Gr\"obner bases and subdues the notorious intermediate expression swell problem and intermediate coefficient swell problem to a substantial extent. The computation of an $S$-polynomial for the new bases requires at most $O(m\ln^2m\ln\ln m)$ word ...
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We consider three modifications of our involutive algorithm for computing Janet bases. These modifications are related to degree compatible monomial orders and specify selection strategies for non-multiplicative prolongations. By using the standard data base of polynomial benchmarks for \Gr bases software we compare the modifications and confront them with Magma that implements Faug\`{e}re's $F_4$ algorithm.