Solving zero-dimensional involutive systems

Zharkov, A. Yu.

doi:10.1007/978-3-0348-9104-2_20

Cited by 13 publications

(9 citation statements)

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“…Its correctness is proved for any continuous involutive division and for arbitrary admissible monomial ordering, while its termination holds, generally, for noetherian divisions. The algorithm is an improved and generalized version of one proposed in [14,15], and has been implemented in Reduce for Pommaret division. The main improvement is the incorporation of Buchberger's chain criterion [16].…”

Section: Introductionmentioning

confidence: 99%

Involutive bases of polynomial ideals

Gerdt¹,

Blinkov²

1998

Mathematics and Computers in Simulation

158

228

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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.

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Section: Introductionmentioning

confidence: 99%

Involutive bases of polynomial ideals

Gerdt¹,

Blinkov²

1998

Mathematics and Computers in Simulation

158

228

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“…, deg k−1 (g) ] appearing in Definition 2. Hence, x j ∈ N M J (f, G) in contradiction to (8). Therefore, deg(w) ≥ 2 and (8) can be rewritten as…”

Section: Relation Between Polynomial Pommaret and Janet Basesmentioning

confidence: 95%

“…Pommaret bases of homogeneous ideals in generic position coincide with their reduced Gröbner bases [7]. The use of these bases makes more accessible the structural information of zero-dimensional ideals [8]. Pommaret bases provide an algorithmic tool for determining combinatorial decompositions of polynomial modules and for computations in the syzygy modules [9].…”

Section: Introductionmentioning

confidence: 99%

On the Relation between Pommaret and Janet Bases

Gerdt

2000

Computer Algebra in Scientific Computing

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In this paper the relation between Pommaret and Janet bases of polynomial ideals is studied. It is proved that if an ideal has a finite Pommaret basis then the latter is a minimal Janet basis. An improved version of the related algorithm for computation of Janet bases, initially designed by Zharkov, is described. For an ideal with a finite Pommaret basis, the algorithm computes this basis. Otherwise, the algorithm computes a Janet basis which need not be minimal. The obtained results are generalized to linear differential ideals.

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“…In the commutative case, it is strictly related with the notion of Janet basis introduced by Zharkov (1996) and the papers cited there and also Apel (1998), as a generalization of Janet's (1929) theory of partial differential equations.…”

Section: Proposition 1 (Properties Of the Semigroup Ideal Regions)mentioning

confidence: 96%