Involutive Bases Research Articles

Recursive structures in involutive bases theory

We study characterisations of involutive bases using a recursion over the variables in the underlying polynomial ring and corresponding completion algorithms. Three key ingredients are (i) an old result by Janet recursively characterising Janet bases for which we provide a new and simpler proof, (ii) the Berkesch–Schreyer variant of Buchberger's algorithm and (iii) a tree representation of sets of terms also known as Janet trees. We start by extending Janet's result to a recursive criterion for minimal Janet bases leading to an algorithm to minimise any given Janet basis. We then extend Janet's result also to Janet-like bases as introduced by Gerdt and Blinkov. Next, we design a novel recursive completion algorithm for Janet bases. We study then the extension of these results to Pommaret bases. It yields a novel recursive characterisation of quasi-stability which we use for deterministically constructing “good” coordinates more efficiently than in previous works. A small modification leads to a novel deterministic algorithm for putting an ideal into Nœther position. Finally, we provide a general theory of involutive-like bases with special emphasis on Pommaret-like bases and study the syzygy theory of Janet-like and Pommaret-like bases.

Complementary decompositions of monomial ideals and involutive bases

Complementary decompositions of monomial ideals—also known as Stanley decompositions—play an important role in many places in commutative algebra. In this article, we discuss and compare several algorithms for their computation. This includes a classical recursive one, an algorithm already proposed by Janet and a construction proposed by Hironaka in his work on idealistic exponents. We relate Janet’s algorithm to the Janet tree of the Janet basis and extend this idea to Janet-like bases to obtain an optimised algorithm. We show that Hironaka’s construction terminates, if and only if the monomial ideal is quasi-stable. Furthermore, we show that in this case the algorithm of Janet determines the same decomposition more efficiently. Finally, we briefly discuss how these results can be used for the computation of primary and irreducible decompositions.

Cellular reductions of the Pommaret-Seiler resolution for Quasi-stable ideals

Involutive bases were introduced in [6] as a type of Gröbner bases with additional combinatorial properties. Pommaret bases are a particular kind of involutive bases with strong relations to commutative algebra and algebraic geometry[11, 12].

An Involutive GVW Algorithm and the Computation of Pommaret Bases

The GVW algorithm computes simultaneously Gröbner bases of a given ideal and of the syzygy module of the given generating set. In this work, we discuss an extension of it to involutive bases. Pommaret bases play here a special role in several respects. We distinguish between a fully involutive GVW algorithm which determines involutive bases for both the given ideal and the syzygy module and a semi-involutive version which computes for the syzygy module only an ordinary Gröbner basis. A prototype implementation of the developed algorithms in Maple is described.

Relative Gröbner and Involutive Bases for Ideals in Quotient Rings

Relative Gröbner and Involutive Bases for Ideals in Quotient Rings

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.

Degree Upper Bounds for Involutive Bases

The aim of this paper is to investigate upper bounds for the maximum degree of the elements of any minimal Janet basis of an ideal generated by a set of homogeneous polynomials. The presented bounds depend on the number of variables and the maximum degree of the generating set of the ideal. For this purpose, by giving a deeper analysis of the method due to Dube (SIAM J Comput 19:750–773, 1990), we improve (and correct) his bound on the degrees of the elements of a reduced Grobner basis. By giving a simple proof, it is shown that this new bound is valid for Pommaret bases, as well. Furthermore, based on Dube’s method, and by introducing two new notions of genericity, so-called J-stable position and prime position, we show that Dube’s (new) bound holds also for the maximum degree of polynomials in any minimal Janet basis of a homogeneous ideal in any of these positions. Finally, we study the introduced generic positions by proposing deterministic algorithms to transform any given homogeneous ideal into these positions.

Computation of Involutive and Gröbner Bases Using the Tableau Representation of Polynomials

For the work with polynomials such data representations as lists of terms, geobuckets, and heaps are usually used. In this paper, an attempt to give a new look on the representation of polynomials for computing involutive and Grobner bases of systems of nonlinear polynomial equations is made. The proposed approach makes it possible to execute a part of this computational task on the GPU, which opens prospects for solving more complicated problems.

Computing Gröbner and Involutive Bases for Linear Systems of Difference Equations

The computation of involutive bases and Gröbner bases for linear systems of difference equations is solved and its importance for physical and mathematical problems is discussed. The algorithm and issues concerning its implementation in C are presented and calculation times are compared with the competing programs. The paper ends with consideration on the parallel version of this implementation and its scalability.

Resolutions and Betti Numbers of Polynomial Modules via Involutive Bases

AbstractWe describe a novel approach to the computation of free resolutions and of Betti numbers of polynomial modules based on a combination of the theory of involutive bases with algebraic discrete Morse theory. This approach allows for the first time to compute Betti numbers (even single ones) without determining a whole resolution which in many cases drastically reduces the computation time. (© 2016 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

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

In the computation of involutive and Grobner 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 Grobner 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.

Term-ordering free involutive bases

In this paper, we consider a monomial ideal J◃P:=A[x1,…,xn], over a commutative ring A, and we face the problem of the characterization for the family Mf(J) of all homogeneous ideals I◃P such that the A-module P/I is free with basis given by the set of terms in the Gröbner escalier N(J) of J. This family is in general wider than that of the ideals having J as initial ideal w.r.t. any term-ordering, hence more suited to a computational approach to the study of Hilbert schemes.For this purpose, we exploit and enhance the concepts of multiplicative variables, complete sets and involutive bases introduced by Riquier (1893, 1899, 1910) and in Janet (1920, 1924, 1927) and we generalize the construction of J-marked bases and term-ordering free reduction process introduced and deeply studied in Bertone et al. (2013a), Cioffi and Roggero (2011) for the special case of a strongly stable monomial ideal J.Here, we introduce and characterize for every monomial ideal J a particular complete set of generators F(J), called stably complete, that allows an explicit description of the family Mf(J). We obtain stronger results if J is quasi-stable, proving that F(J) is a Pommaret basis and Mf(J) has a natural structure of affine scheme.The final section presents a detailed analysis of the origin and the historical evolution of the main notions we refer to.

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.

Parallel modular computation of Gröbner and involutive bases

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

BIBasis, a package for reduce and Macaulay2 computer algebra systems to compute Boolean involutive and Gröbner bases

In this paper, we describe the BIBasis package designed for REDUCE and Macaulay2 computer algebra systems, which allows one to compute Boolean involutive bases and Grobner bases. The implementations and user interfaces of the package for both systems are described in the respective sections of the paper. Also, we present results of comparisons of BIBasis with other packages and algorithms for constructing Boolean Grobner bases available in the computer algebra systems.

On computation of Boolean involutive bases

Grobner bases in Boolean rings can be calculated by an involutive algorithm based on the Janet or Pommaret division. The Pommaret division allows calculations immediately in the Boolean ring, whereas the Janet division implies use of a polynomial ring over field $$ \mathbb{F}_2 $$ . In this paper, both divisions are considered, and distributive and recursive representations of Boolean polynomials are compared from the point of view of calculation effectiveness. Results of computer experiments with both representations for an algorithm based on the Pommaret division and for lexicographical monomial order are presented.

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.

A combinatorial approach to involution and δ-regularity II: structure analysis of polynomial modules with pommaret bases

Much of the existing literature on involutive bases concentrates on their efficient algorithmic construction. By contrast, we are here more concerned with their structural properties. Pommaret bases are particularly useful in this respect. We show how they may be applied for determining the Krull and the projective dimension, respectively, and the depth of a polynomial module. We use these results for simple proofs of Hironaka’s criterion for Cohen–Macaulay modules and of the graded form of the Auslander–Buchsbaum formula, respectively. Special emphasis is put on the syzygy theory of Pommaret bases and its use for the construction of a free resolution which is generically minimal for componentwise linear modules. In the monomial case, the arising complex always possesses the structure of a differential algebra and it is possible to derive an explicit formula for the differential. Here a minimal resolution is obtained, if and only if a stable module is treated. These observations generalise results by Eliahou and Kervaire. Using our resolution, we show that the degree of the Pommaret basis with respect to the degree reverse lexicographic term order is always the Castelnuovo–Mumford regularity. This approach leads to new proofs for a number of characterisations of this invariant proposed in the literature. This includes in particular the criteria of Bayer/Stillman and Eisenbud/Goto, respectively. We also relate Pommaret bases to the recent work of Bermejo/Gimenez and Trung on computing the Castelnuovo–Mumford regularity via saturations. It is well-known that Pommaret bases do not always exist but only in so-called δ-regular coordinates. We show that several classical results in commutative algebra, holding only generically, are true for these special coordinates. In particular, they are related to regular sequences, independent sets of variables, saturations and Noether normalisations. Many properties of the generic initial ideal hold also for the leading ideal of the Pommaret basis with respect to the degree reverse lexicographic term order, although the latter one is in general not Borel-fixed. We present a deterministic approach for the effective construction of δ-regular coordinates that is more efficient than all methods proposed in the literature so far.

Efficiency estimate for distributed computation of Gröbner bases and involutive bases

Several years ago, we presented a program complex for parallel computation of Grobner 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.

Specialized computer algebra system GINV

Computer algebra system GINV (Grobner INVolutive) is presented. It is designed for studying and solving systems of algebraic, differential, and difference equations of polynomial type by their completion to involution. The system is based on algorithms for constructing involutive Janet and Janet-like bases for polynomial ideals and modules, as well as reduced Grobner bases, developed by the authors. GINV consists of a library of C++ programs, which is a module of the Python language. It is available at http://invo.jinr.ru/ginv/ and is distributed under the terms of the GPL v2.