Involutive Basis Research Articles

Involutive bases algorithm incorporating F5 criterion

Faugèreʼs F5 algorithm (Faugère, 2002) is the fastest known algorithm to compute Gröbner bases. It has a signature-based and an incremental structure that allow to apply the F5 criterion for deletion of unnecessary reductions. In this paper, we present an involutive completion algorithm which outputs a minimal involutive basis. Our completion algorithm has a non-incremental structure and in addition to the involutive form of Buchbergerʼs criteria it applies the F5 criterion whenever this criterion is applicable in the course of completion to involution. In doing so, we use the G2V form of the F5 criterion developed by Gao, Guan and Volny IV (Gao et al., 2010a). To compare the proposed algorithm, via a set of benchmarks, with the Gerdt–Blinkov involutive algorithm (Gerdt and Blinkov, 1998) (which does not apply the F5 criterion) we use implementations of both algorithms done on the same platform in Maple.

A combinatorial approach to involution and δ-regularity I: involutive bases in polynomial algebras of solvable type

Involutive bases are a special form of non-reduced Grobner bases with additional combinatorial properties. Their origin lies in the Janet–Riquier theory of linear systems of partial differential equations. We study them for a rather general class of polynomial algebras including also non-commutative algebras like those generated by linear differential and difference operators or universal enveloping algebras of (finite-dimensional) Lie algebras. We review their basic properties using the novel concept of a weak involutive basis and present concrete algorithms for their construction. As new original results, we develop a theory for involutive bases with respect to semigroup orders (as they appear in local computations) and over coefficient rings, respectively. In both cases it turns out that generally only weak involutive bases exist.

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.

Complete involutive rewriting systems

Given a monoid string rewriting system M , one way of obtaining a complete rewriting system for M is to use the classical Knuth–Bendix critical pairs completion algorithm. It is well-known that this algorithm is equivalent to computing a noncommutative Gröbner basis for M . This article develops an alternative approach, using noncommutative involutive basis methods to obtain a complete involutive rewriting system for M .

A C# package for assembling quantum circuits and generating associated polynomial sets

Recently it has been shown that elements of the unitary matrix determined by a quantum circuit can be computed by counting the number of common roots in the finite field ℤ2 for a certain set of multivariate polynomials over ℤ2. In this paper we present a C# package that allows a user to assemble a quantum circuit and to generate the multivariate polynomial set associated with the circuit. The generated polynomial system can further be converted to the canonical triangular involutive basis form, which is appropriate for counting the number of common roots of the polynomials.

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.

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.

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 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.

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.

Gröbner and involutive bases for zero-dimensional ideals

A new characterization of involutive systems is presented, which leads to a connection between Gr&amp;ouml;bner basis and involutive systems, and allows to construct, in an easy way, an involutive basis from a Gr&amp;ouml;bner basis.