Involutive Divisions Research Articles

Applications of Bar Code to Involutive Divisions and a “Greedy” Algorithm for Complete Sets

Given a finite set of terms U in n variables, we describe an algorithm which finds – if it exists – an ordering on the variables such that U is a complete set according to Janet involutive division. The algorithm, based on Bar Codes for monomial ideals, is able to adjust the variables ordering with a sort of backtracking technique, thus allowing to find the desired ordering without trying all the n! possible ones.

Toward involutive bases over effective rings

In this paper we extend the theory of involutive divisions to the case of monomials with coefficients over effective rings. Moreover, as regards involutive bases, we study the computation of weak involutive bases and sketch a conjecture on strong involutive bases.

Involutive divisions and monomial orderings: Part II

This paper is a sequel to the studies on classification properties of involutive divisions reported in [1]. An example is given in which the minimal involutive basis of a particular monomial ideal for the "Janet antipode" <-division is neither an involutive Janet basis nor a minimal basis for a Janet-like division for any of n! orderings of variables. This example disproves the hypothesis that the minimal involutive basis for continuous and constructive divisions always coincides with the Janet basis for some ordering of variables.

Involutive divisions and monomial orderings

In the paper, two classes of involutive divisions related to admissible monomial orderings--?-divisions and ?-divisions--are considered. The latter may be viewed as an improvement of the class of induced divisions introduced in [1]. The continuity and constructivity of the ?-divisions, as well as a number of additional properties of the ?-divisions, are proved. Taking into account that the Janet division is a ?-division associated with the lexicographical order, its "antipode"--a ?-division associated with the same order--is separated in the class of the ?-divisions.

Involutive divisions for effective involutive algorithms

Properties of involutive divisions on monomials are studied. A new method of involutive graphs is developed. The concept of complete global involutive division is introduced. A criterion of Noetherianity of involutive divisions, a property of graphs of global involutive division, a test for completeness of global involutive division, and a criterion of global involutive division are considered. A new series of involutive divisions is obtained by the process of completion. The properties of the divisions contained in the constructed series are studied. It is shown that the divisions from the series are better than the classical involutive divisions for involutive algorithms. A problem stated by Gao is solved: another series of involutive divisions is obtained. It is proved that all divisions of this series are continuous.

Pair analysis of involutive divisions

The main goal of this work is to describe a new approach to the study of involutive divisions using the pairwise property. The paper presents a simple and intuitive method for constructing the Janet division and reveals the deep intrinsic relationship between Janet division and Lex-ordering. A method for constructing some analogues of the Janet division for other orders is described. An example of pairwise, continuous, and nonconstructive involutive division is given.

On constructivity of involutive divisions

In the paper, the problem of existence of the minimal involutive monomial basis is considered for different involutive divisions. The existence criterion is obtained in the form of a constructivity property. In the paper, new examples of constructive and nonconstructive (> ?)-divisions are presented.

Involutive divisions: Slice and pair properties

Two classes of involutive divisions are considered: pairwise and sliced ones. Pairwise involutive divisions are convenient for the calculation of multiplicative and nonmultiplicative variables when new monomials are added. Directed sliced partitions are convenient for finding involutive divisors of an arbitrary monomial. It is proved that the only homothetic division that is simultaneously pairwise and directed sliced is the Janet involutive division. This shows that the Janet division is optimal for the calculation of multiplicative variables and for the fast search of a divisor.

Involutive Divisions. Graphs

A new approach to the theory of involutive divisions is suggested. The application of this approach simplifies the formulation of definitions and properties and makes it possible to obtain new involutive divisions. The latter feature is important in light of the results discussed in l1r. An involutive division graph is defined, and the notions of its completeness and projection are introduced. Criteria for being global and Noetherian and a sufficient condition for the completeness are obtained.

Involutive Division Techniques: Some Generalizations and Optimizations

A new class of involutive divisions induced by certain orderings of monomials is considered. It is proved that these divisions are Noetherian and constructive. Therefore, each of them allows one to compute an involutive Grobner basis of a polynomial ideal by sequentially examining multiplicative reductions of nonmultiplicative prolongations. The dependence of involutive algorithms on the completion ordering is studied. Based on the properties of particular involutive divisions, two computational optimizations are suggested. One of them consists of a special choice of the completion ordering. The other optimization is related to recomputing multiplicative and nonmultiplicative variables in the course of the algorithm. Bibliography: 17 titles.

Involutive directions and new involutive divisions

In this paper, we propose the concept of involutive direction as a vector representation for the concept of involutive division proposed by Gerdt and his co-workers. With this representation, most of the properties of involutive divisions such as Noetherity, Artinity, and constructivity, can be greatly simplified. A new algorithm to compute the involutive completion is also given. Based on the vector representation, two new types of involutive divisions are found and proved to be Noetherian, Artinian, and constructive. These new divisions may lead to new methods of finding integrability conditions of partial differential equations and computing Gröbner bases of polynomial ideals.

Method of Separative Monomials for Involutive Divisions

A method of separative monomials is presented for constructing a partition of a set of monomials associated with a certain involutive division. Given a particular set, the method is capable of constructing a search tree of an involutive divisor with various balancing criteria.

The Theory of Involutive Divisions and an Application to Hilbert Function Computations

Generalising the divisibility relation of terms we introduce the lattice of so-called involutive divisions and define the admissibility of such an involutive division for a given set of terms. Based on this theory we present a new approach for building a general theory of involutive bases of polynomial ideals. In particular, we give algorithms for checking the involutive basis property and for completing an arbitrary basis to an involutive one. It turns out that our theory is more constructive and more flexible than the axiomatic approach to general involutive bases due to Gerdt and Blinkov.Finally, we show that an involutive basis contains more structural information about the ideal of leading terms than a Gröbner basis and that it is straightforward to compute the (affine) Hilbert function of an idealIfrom an arbitrary involutive basis of alI.

Minimal involutive bases

Abstract In this paper, we present an algorithm for construction of minimal involutive polynomial bases which are Grobner 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 Grobner basis.

Involutive bases of polynomial ideals

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 separation is applied to polynomials in terms of their leading monomials. As special cases of the separation we consider those introduced by Janet, Thomas and Pommaret for the purpose of algebraic analysis of partial differential equations. Given involutive division, we define an involutive reduction and an involutive normal form. Then we introduce, in terms of the latter, the concept of involutivity for polynomial systems. We prove that an involutive system is a special, generally redundant, form of a Gröbner basis. An algorithm for construction of involutive bases is proposed. It is shown that involutive divisions satisfying certain conditions, for example, those of Janet and Thomas, provide an algorithmic construction of an involutive basis for any polynomial ideal. Some optimization in computation of involutive bases is also analyzed. In particular, we incorporate Buchberger's chain criterion to avoid unnecessary reductions. The implementation for Pommaret division has been done in Reduce.

Minimal involutive bases

In this paper, we present an algorithm for construction of minimal involutive polynomial bases which are Grobner 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 Grobner basis. # 1998 IMACS/Elsevier Science B.V.