Effectiveness of involutive criteria in computation of polynomial Janet bases

Gerdt, Vladimir P.; Yanovich, D. A.

doi:10.1134/s0361768806030030

Cited by 7 publications

(3 citation statements)

References 6 publications

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“…In the differential case, application of criteria can decrease computation time by avoiding useless reductions of non-reductive prolongations. Janet's combinatorial approach already avoids many reductions of ∆-polynomials, as used in other approaches (see Gerdt and Yanovich (2006)). In addition, we use the involutive criteria 2-4 (cf.…”

Section: Algorithmic Optimizationsmentioning

confidence: 99%

Algorithmic Thomas decomposition of algebraic and differential systems

Bächler

Gerdt

Lange-Hegermann

et al. 2012

Journal of Symbolic Computation

100

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In this paper, we consider systems of algebraic and non-linear partial differential equations and inequations. We decompose these systems into so-called simple subsystems and thereby partition the set of solutions. For algebraic systems, simplicity means triangularity, square-freeness and nonvanishing initials. Differential simplicity extends algebraic simplicity with involutivity. We build upon the constructive ideas of J. M. Thomas and develop them into a new algorithm for disjoint decomposition. The given paper is a revised version of Bächler et al. (2010) and includes the proofs of correctness and termination of our decomposition algorithm. In addition, we illustrate the algorithm with further instructive examples and describe its Maple implementation together with an experimental comparison to some other triangular decomposition algorithms.Keywords: disjoint triangular decomposition, simple systems, polynomial systems, differential systems, involutivity input by means of regular chains (if the input only consists of equations) or regular systems. However, the Thomas decomposition differs noticeably from this decomposition, since the Thomas decomposition is finer and demands disjointness of the solution set. For a detailed description of algorithms related to regular chains, we refer the reader to Moreno Maza (1999).The disjointness of the Thomas decomposition combined with the structural properties of simple systems provide a useful platform for counting solutions of polynomial systems. In fact, the Thomas decomposition is the only known method to compute the counting polynomial introduced by Plesken (2009a). We refer to §2.3 for details on this structure, counting and their applications.During his research on triangular decomposition, Thomas was motivated by the Riquier-Janet theory (cf. Riquier (1910); Janet (1929)), extending it to non-linear systems of partial differential equations. For this purpose he developed a theory of (Thomas) monomials, which generate an involutive monomial division nowadays called Thomas division (cf. Gerdt and Blinkov (1998a)). He gave a recipe for decomposing a non-linear differential system into algebraically simple and passive subsystems (cf. Thomas (1937)). A modified version of the differential Thomas decomposition was considered by Gerdt (2008) with its link to the theory of involutive bases (cf. Gerdt and Blinkov (1998a);Gerdt (2005Gerdt ( , 1999; Seiler (2010)). In this decomposition, the output systems are Janet-involutive in accordance to the involutivity criterion from Gerdt (2008) and hence they are coherent. For a linear differential system it is a Janet basis of the corresponding differential ideal, as computed by the Maple package Janet (cf. Blinkov et al. (2003)).The differential Thomas decomposition differs from that computed by the Rosenfeld-Gröbner algorithm (cf. Boulier et al. (2009, 1995)). The latter decomposition forms a basis of the diffalg, DifferentialAlgebra and BLAD packages (cf. Hubert (1996-2004); Boulier (2004Boulier ( -2009). Experimentally, we f...

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Section: Algorithmic Optimizationsmentioning

confidence: 99%

Algorithmic Thomas decomposition of algebraic and differential systems

Bächler

Gerdt

Lange-Hegermann

et al. 2012

Journal of Symbolic Computation

100

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“…Moreover, this subalgorithm detects some unnecessary reductions using the involutive form of Buchberger's criteria. It is worth noting that the first author and Yanovich [GY06] have shown that the use of C 3 and C 4 criteria (see Proposition 8) may notably slow down the computation of involutive bases. That is why, we use only C 1 and C 2 criteria.…”

Section: Algorithm 6 Vargerdtmentioning

confidence: 99%

A Variant of Gerdt's Algorithm for Computing Involutive Bases

Gerdt¹,

Hashemi²,

-Alizadeh³

2011

Preprint

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In [Ger05], the first author presented an efficient algorithm for computing involutive (and reduced Gröbner) 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 with its original form described in [Ger05] and has been implemented in Maple too. In doing so, we have taken care to provide uniform implementation details for the both algorithms.

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“…Then, Apel and Hemmecke in [2] discovered two more criteria (see also [18]) that in the aggregate with C 2 are equivalent to Buchberger's chain criterion. The computer experimentation done by the first author and Yanovich [23] revealed that these two criteria, being applied when the criteria C 1 and C 2 are not applicable, often (for not very large examples) slowdown computation of involutive bases. That is why, in the given paper we use only the criteria C 1 and C 2 .…”

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confidence: 99%

Involutive bases algorithm incorporating F5 criterion

Gerdt¹,

Hashemi²,

M.-Alizadeh³

2013

Journal of Symbolic Computation

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Faugère's F5 algorithm [13] 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 nonincremental structure and in addition to the involutive form of Buchberger's criteria it applies the F5 criterion whenever this criterion is applicable in the course of completion to involution. In doing so, we use the G 2 V form of the F5 criterion developed by Gao, Guan and Volny IV [14]. To compare the proposed algorithm, via a set of benchmarks, with the Gerdt-Blinkov involutive algorithm [19] (which does not apply the F5 criterion) we use implementations of both algorithms done on the same platform in Maple.

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Effectiveness of involutive criteria in computation of polynomial Janet bases

Cited by 7 publications

References 6 publications

Algorithmic Thomas decomposition of algebraic and differential systems

Algorithmic Thomas decomposition of algebraic and differential systems

A Variant of Gerdt's Algorithm for Computing Involutive Bases

Involutive bases algorithm incorporating F5 criterion

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