Non-commutative Gröbner bases in algebras of solvable type

Kandri-Rody, Abdelilah; Weispfenning, Volker

doi:10.1016/s0747-7171(08)80003-x

Cited by 195 publications

(170 citation statements)

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“…If we want to generate a derivative rule from (29), this is an elimination problem in the given noncommutative polynomial ring which can be solved by Grobner basis methods ( [ 5 ] , [13], [36], [39], [31]- [33]). The Grobner basis of the left ideal generated by (29) with respect to the lexicographic term order ( D , N , n , x ) is given by we used the REDUCE implementation [24] (see [12]) for the noncommutative Grobner calculations of this article.…”

Section: (~~-1 )~~+ 2 X~-n ( L + N ) = O and ( N + 2 ) N 2 -( 3 + 2 Nmentioning

confidence: 99%

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Identities for families of orthogonal polynomials and special functions

Koepf

1997

Integral Transforms and Special Functions

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In this article we present new results for families of orthogonal polynomials and special functions, that are determined by algorithmical approaches. In the first section, we present new results, especially for discrete families of orthogonal polynomials, obtained by an application of the celebrated Zeilberger algorithm. Next, we present algorithms for holonomic families f (n, x) of special functions which possess a derivative rule. We call those families admissible. A family f (n,x) is holonomic if it satisfies a holonomic recurrence equation with respect to n, and a holonomic differential equation with respect to z , i. e. linear homogeneous equations with polynomial coefficients. The rather rigid property of admissibility has many interesting consequences, that can be used to generate and verify identities for these functions by linear algebra techniques. On the other hand, many families of special functions, in particular families of orthogonal polynomials, are admissible. We moreover present a method that generates the derivative rule from the h o b nomic representation of a holonomic family. As examples, we find new identities for the Jacobi polynomials and for the Whittaker functions, and for families of discrete orthogonal polynomials by the given approach. Finally, we present representations for the parameter derivatives of the Gegenbauer and the generalized Laguerre polynomials.

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Section: (~~-1 )~~+ 2 X~-n ( L + N ) = O and ( N + 2 ) N 2 -( 3 + 2 Nmentioning

confidence: 99%

“…(Admissible family of special functions) We call a family fn of functions admissibleif they satisfy a recurrence equation of type (1) and a derivative rule of type (13). We call the order of the recurrence equation the order of the admissible family fn .…”

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confidence: 99%

Identities for families of orthogonal polynomials and special functions

Koepf

1997

Integral Transforms and Special Functions

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“…All these reductions satisfy the conditions of Definition 2.5 and therefore are Gröbner reductions. Note that the reduction with respect to several orderings defined in [16], which is a special instance of the Gröbner reduction, can be naturally applied to algebras of a certain subclass of the class of algebras of solvable type (algebras of solvable type were introduced and studied in [11]). This subclass consists of algebras of solvable type R = K {X 1 , .…”

Section: Definition 25 (Gröbner Reduction)mentioning

confidence: 99%

Relative Reduction and Buchberger’s Algorithm in Filtered Free Modules

Fürst

Levin

2017

Math.Comput.Sci.

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In this paper we develop a relative Gröbner basis method for a wide class of filtered modules. Our general setting covers the cases of modules over rings of differential, difference, inversive difference and differencedifferential operators, Weyl algebras and multiparameter twisted Weyl algebras (the last class of rings includes the classes of quantized Weyl algebras and twisted generalized Weyl algebras). In particular, we obtain a Buchbergertype algorithm for constructing relative Gröbner bases of filtered free modules.

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“…A is a solvable polynomial algebra [14]. Any solvable polynomial algebra can be viewed as a skew solvable polynomial ring in this way.…”

Section: Skew Solvable Polynomial Ringsmentioning

confidence: 99%

A signature-based algorithm for computing Gröbner-Shirshov bases in skew solvable polynomial rings

Zhao

Zhang

2015

Open Mathematics

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Signature-based algorithms are efficient algorithms for computing Gröbner-Shirshov bases in commutative polynomial rings, and some noncommutative rings. In this paper, we first define skew solvable polynomial rings, which are generalizations of solvable polynomial algebras and (skew) PBW extensions. Then we present a signature-based algorithm for computing Gröbner-Shirshov bases in skew solvable polynomial rings over fields.

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Non-commutative Gröbner bases in algebras of solvable type

Cited by 195 publications

References 10 publications

Identities for families of orthogonal polynomials and special functions

Identities for families of orthogonal polynomials and special functions

Relative Reduction and Buchberger’s Algorithm in Filtered Free Modules

A signature-based algorithm for computing Gröbner-Shirshov bases in skew solvable polynomial rings

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