Involutive Bases Research Articles

On the homogeneous Gröbner basis for tensors

Algorithmic methods of commutative algebra based on the involutive and Grobner bases technique are efficient means for completion of equations governing dynamical systems to involution. At the same time, when working with high-dimensional tensor quantities, direct use of standard functions for calculating Grobner bases, which are built in computer algebra systems Maple and Mathematica, requires much memory. However, being multilinear forms, tensors admit special grading that makes it possible to classify polynomials in terms of their degree of homogeneity. With regard to this feature, we propose to use a special homogeneous Grobner basis, which allows us to avoid difficulties associated with large amount of computation. Such a basis is constructed step by step, as the degree of the polynomial grows. As an example, an algorithm for constructing the homogeneous basis in a finite-dimensional Hamiltonian system with many polynomial constraints (the so-called Yang-Mills mechanics) is presented.

Asymmetric approach to computation of Gröbner bases

A new approach to Buchberger’s algorithm based on the use of essential multiplications and nonmultiplicative prolongations instead of traditional S-polynomials is described. In the framework of this approach, both Buchberger’s algorithm for computing Gröbner bases and the Gerdt-Blinkov algorithm for computing involutive bases obtain a unified form of description. The new approach is based on consideration of the process of determining an S-polynomial as a process of constructing a nonmultiplicative prolongation of a polynomial and its subsequent reducing with respect to an essential multiplication. An advantage of the method is that some “redundant” S-pairs are automatically excluded from consideration.

Involutive Bases Under Composition

In this paper, the behaviors of involutive bases under composition operation are studied. For two kinds of involutive bases, i.e., Pommaret bases, Janet bases, we study their behavior problems under composition. Some further problems are also proposed.

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.

Detecting unnecessary reductions in an involutive basis computation

We consider the check of the involutive basis property in a polynomial context. In order to show that a finite generating set F of a polynomial ideal I is an involutive basis one must confirm two properties. Firstly, the set of leading terms of the elements of F has to be complete. Secondly, one has to prove that F is a Gröbner basis of I . The latter is the time critical part but can be accelerated by application of Buchberger’s criteria including the many improvements found during the last two decades. Gerdt and Blinkov [Gerdt, V.P., Blinkov, Y.A., 1998. Involutive bases of polynomial ideals. Mathematics and Computers in Simulation 45, 519–541] were the first who applied these criteria in involutive basis computations. We present criteria which are also transferred from the theory of Gröbner bases to involutive basis computations. We illustrate that our results exploit the Gröbner basis theory slightly more than those of Gerdt and Blinkov. Our criteria apply in all cases where those of Gerdt/Blinkov do, but we also present examples where our criteria are superior. Some of our criteria can also be used in algebras of solvable type, e.g., Weyl algebras or enveloping algebras of Lie algebras, in full analogy to the Gröbner basis case. We show that the application of criteria enforces the termination of the involutive basis algorithm independent of the prolongation selection strategy.

On a Conjecture of R. P. Stanley; Part I—Monomial Ideals

In 1982 Richard P. Stanley conjectured that any finitely generated \Bbb Zn-graded module M over a finitely generated \Bbb Nn-graded \Bbb K-algebra R can be decomposed in a direct sum M e ⊕i e 1t νiSi of finitely many free modules νiSi which have to satisfy some additional conditions. Besides homogeneity conditions the most important restriction is that the Si have to be subalgebras of R of dimension at least depth M. We will study this conjecture for the special case that R is a polynomial ring and M an ideal of R, where we encounter a strong connection to generalized involutive bases. We will derive a criterion which allows us to extract an upper bound on depth M from particular involutive bases. As a corollary we obtain that any monomial ideal M which possesses an involutive basis of this type satisfies Stanley's Conjecture and in this case the involutive decomposition defined by the basis is also a Stanley decomposition of M. Moreover, we will show that the criterion applies, for instance, to any monomial ideal of depth at most 2, to any monomial ideal in at most 3 variables, and to any monomial ideal which is generic with respect to one variable. The theory of involutive bases provides us with the algorithmic part for the computation of Stanley decompositions in these situations.

Involutive Bases in the Weyl Algebra

We discuss the definition and the construction of involutive bases, a special kind of Gröbner bases, for left ideals in the Weyl algebra. We consider not only term orders, for which the extension is straightforward, but also the more general case of multiplicative monomial orders.

On an Algorithmic Optimization in Computation of Involutive Bases

An optimization of a general algorithm for computation of the minimal involutive basis is suggested. It can be applied for an arbitrary constructive involutive division. The optimization suggested minimizes the number of redistributions between elements of an intermediate basis required in the process of computations

Involutive bases of ideals in the ring of polynomials

This paper continues the study of the relationship between Grobner bases and involutive bases of polynomial ideals started in [1, 5]. Involutive bases are considered as Grobner bases with some fixed normal form algorithms. It is shown that once the search order of critical pairs is coordinated, the sameS-elements and their reductions appear in the same order in the process of the construction of Grobner bases and involutive bases.

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.

A Gröbner Approach to Involutive Bases

Recently, Zharkov and Blinkov introduced the notion of involutive bases of polynomial ideals. This involutive approach has its origin in the theory of partial differential equations and is a translation of results of Janet and Pommaret. In this paper we present a pure algebraic foundation of involutive bases of Pommaret type. In fact, they turn out to be generalized left Gröbner bases of ideals in the commutative polynomial ring with respect to a non-commutative grading. The introduced theory will allow not only the verification of the results of Zharkov and Blinkov but it will also provide some new facts.

Gröbner and involutive bases for zero-dimensional ideals

A new characterization of involutive systems is presented, which leads to a connection between Gröbner basis and involutive systems, and allows to construct, in an easy way, an involutive basis from a Gröbner basis.