The Computation of Gröbner Bases Using an Alternative Algorithm

Apel, Joachim

doi:10.1007/978-3-0348-8800-4_2

Cited by 3 publications

(1 citation statement)

References 7 publications

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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%

Computing Gröbner Bases by FGLM Techniques in a Non-commutative Setting

Borges-Trenard

Borges-Quintana

Mora

2000

Journal of Symbolic Computation

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

confidence: 96%

Computing Gröbner Bases by FGLM Techniques in a Non-commutative Setting

Borges-Trenard

Borges-Quintana

Mora

2000

Journal of Symbolic Computation

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The Theory of Involutive Divisions and an Application to Hilbert Function Computations

Apel

1998

Journal of Symbolic Computation

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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 ideal I from an arbitrary involutive basis of I.

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

Seiler

2009

AAECC

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Involutive bases are a special form of non-reduced Gröbner 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 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.

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The Computation of Gröbner Bases Using an Alternative Algorithm

Cited by 3 publications

References 7 publications

Computing Gröbner Bases by FGLM Techniques in a Non-commutative Setting

Computing Gröbner Bases by FGLM Techniques in a Non-commutative Setting

The Theory of Involutive Divisions and an Application to Hilbert Function Computations

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

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