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Computation of Non-Commutative Gröbner Bases in Grassmann and Clifford Algebras

2010
*
Advances in Applied Clifford Algebras
*

Mora who already in the 1990s considered

doi:10.1007/s00006-010-0205-0
fatcat:a3ronds35vf6vifhyuks5t6c5u
*a*comprehensive and*algorithmic*approach*to**Gröbner*bases for commutative and non-commutative algebras. It was T. ... Tensor, Clifford and Grassmann algebras belong*to**a*wide class*of*non-commutative algebras that have*a*Poincaré-Birkhoff-Witt (PBW) "monomial"*basis*. ... Thanks are also due*to*Hans Schönemann (University*of*Kaiserslautern) for his help with monomial orders in Plural. ...##
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Index to Volume 41

2006
*
Journal of symbolic computation
*

devoted

doi:10.1016/s0747-7171(06)00119-2
fatcat:n3voc3w2dvex7bo3gopguifv4m
*to*symbolic computation, 255 Bruno Buchberger's PhD thesis 1965: An*algorithm*for finding the*basis*elements*of*the residue class*ring**of**a*zero dimensional polynomial ideal, 475 BUCHBERGER, B ... bases and logarithmic D-modules, 317*Gröbner*bases in function*rings*-*A*guide for introducing reduction relations*to*algebraic structures, 1264*Gröbner*bases*of*Hilbert ideals*of*alternating*groups*, 905 ... solutions*to*Thomas' family*of*Thue equations over imaginary quadratic number fields, 980 HONG, H., KAPUR, D. and PAULE, P., Bruno Buchberger -*A*life devoted*to*symbolic computation, 255 HOŞ TEN, S. and ...##
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Page 3137 of Mathematical Reviews Vol. , Issue 2002E
[page]

2002
*
Mathematical Reviews
*

in

*a*base*of**A*, is*nilpotent*in gr(*A*) then*A*is*nilpotent*. ... Thus the*algorithms*developed rely on earlier re- search devoted*to**Grobner**basis*theory and presented in several publications by Buchberger, Mora, Reinert, Ufnarovskii, Winkler and other authors. ...##
###
On the leading terms ideals of polynomial ideals over a valuation ring

2012
*
Journal of Algebra
*

Let R be

doi:10.1016/j.jalgebra.2011.11.015
fatcat:kvpxyomqkngdraa6776c6ymahy
*a**ring*.*A*term*a*X k (where*a*∈ R and k ∈ N) belongs*to*an ideal*of*R[ X]*of*the form As an immediate consequence, one obtains: Corollary 5. ... This gives*a*negative answer*to*the open question*of*whether if V is*a*valuation*ring*with Krull dimension 1, then for any finitely*generated*ideal I*of*V[X], the leading terms ideal*of*I is also finitely ... Acknowledgment The authors are grateful*to*the anonymous referee for many useful comments and for proposing Proposition 12. ...##
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Page 6150 of Mathematical Reviews Vol. , Issue 2001I
[page]

2001
*
Mathematical Reviews
*

The results on Newton polytopes are proved by first describing the

*rings**of*constants*of*locally*nilpotent*derivations that are ho- mogeneous with respect*to**a*certain grading, and then, for*general*locally ... Let d be*a*nonzero locally*nilpotent*derivation*of*the polynomial*ring*in n variables over*a*field k*of*characteristic zero. ...##
###
A Generalization of Gröbner Basis Algorithms to Polycyclic Group Rings

1998
*
Journal of symbolic computation
*

Using this reduction we present

doi:10.1006/jsco.1997.0165
fatcat:qw5nz5hlqbhylpbsbkfzriv2uy
*a**generalization**of*Buchberger's*Gröbner**basis*method by giving an appropriate definition*of*"*Gröbner*bases" in this setting and by characterizing them using the concepts ... It is well-known that for the integral*group**ring**of**a*polycyclic*group*several decision problems are decidable, in particular the ideal membership problem. ... This concept was used*to*define*a*Noetherian reduction in*group**rings*over finitely*generated**nilpotent**rings*in Madlener and Reinert (1996) and*to**generalize**Gröbner**basis**algorithms*for right and two-sided ...##
###
Characterization of rank two locally nilpotent derivations in dimension three
[article]

2007
*
arXiv
*
pre-print

Together with an

arXiv:0705.2489v1
fatcat:5fgutkj46rgp7b7v4rwwxaju3i
*algorithm*for computing the plinth ideal, this gives*a*method for computing the rank*of**a*locally*nilpotent*derivation in dimension three. ... In this paper we give an*algorithmic*characterization*of*rank two locally*nilpotent*derivations in dimension three. ... Let G be*a**Gröbner**basis**of*J with respect*to*the lexicographic order u 1 ≺ u 2 ≺ u 3 ≺ x ≺ y ≺ z and G 1 = G ∩ K[u 1 , u 2 , u 3 ]. ...##
###
The isomorphism problem for graded algebras and its application to mod-p cohomology rings of small p-groups
[article]

2015
*
arXiv
*
pre-print

The mod-p cohomology

arXiv:1503.04666v1
fatcat:owfi4grsq5ajxneztc2fhkflwm
*ring**of**a*non-trivial finite p-*group*is an infinite dimensional, finitely presented graded unital algebra over the field with p elements, with*generators*in positive degrees. ... As application, we determine all graded isomorphisms between the mod-p cohomology*rings**of*all p-*groups**of*order at most 100. ... Secondly, we apply our*algorithm**to*the mod-p cohomology*rings**of*the p-*groups*with order at most 100. ...##
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A Divide and Conquer Method to Compute Binomial Ideals
[chapter]

2014
*
Lecture Notes in Computer Science
*

*basis*and that

*Gröbner*

*basis*computation is very sensitive

*to*the number

*of*variables in the

*ring*. ... This work is

*a*

*generalization*

*of*the work done by the authors in [12, 13] and is motivated by the fact that any

*algorithm*

*to*compute binomial ideals spends

*a*significant amount

*of*time computing

*Gröbner*... The Buchberger's

*algorithm*

*to*compute

*Gröbner*

*basis*has been adopted

*to*compute pseudo-

*Gröbner*

*basis*in [13,

*Algorithm*4] . ...

##
###
Gröbner Bases over Commutative Rings and Applications to Coding Theory
[chapter]

2009
*
Gröbner Bases, Coding, and Cryptography
*

We discuss and compare the main

doi:10.1007/978-3-540-93806-4_14
fatcat:5o2prs35uzfmtf4ghdp5cfumwu
*algorithms*that may be implemented*to*compute*Gröbner*and (in the case*of**a*chain*ring*) Szekeres-like bases. ... We give*a*survey*of*results and applications relating*to*the theory*of**Gröbner*bases*of*ideals and modules where the coefficient*ring*is*a*finite commutative*ring*. ... The details*of*an*algorithm**to*compute*a**Gröbner**basis**of*M from*a*standard*basis*can be read in [29,*Algorithm*6.3] , and the*general*procedure has been extended for the*ring*case [2,*Algorithm*2]. • ...##
###
Page 4866 of Mathematical Reviews Vol. , Issue 98H
[page]

1998
*
Mathematical Reviews
*

(D-KSRL-C; Kaiserslautern)

*A**generalization**of*Grébner*basis**algorithms**to**nilpotent**group**rings*. (English summary) Appl. Algebra Engrg. Comm. Comput. 8 (1997), no. 2, 103-123. ...*group**rings*using Grébner*basis**algorithms*and reduc- tion.” ...##
###
Using Extended Derksen Ideals in Computational Invariant Theory
[article]

2014
*
arXiv
*
pre-print

The main purpose

arXiv:1310.6851v2
fatcat:infhszd6lnh2ll5zc73lbgmmda
*of*this paper is*to*develop new*algorithms*for computing invariant*rings*in*a**general*setting. ... In particular, we present an*algorithm*for computing invariants*of**a*finite*group*acting on*a*finitely*generated*algebra over*a*Euclidean*ring*. ... Only her remarks led me*to*think*of*applying the ideas*to*actions*of*finite*groups*on algebras over*rings*rather than fields. ...##
###
Page 5538 of Mathematical Reviews Vol. , Issue 2002H
[page]

2002
*
Mathematical Reviews
*

“The new

*algorithm*described in this article can be used*to*compute*generators*up*to**a*certain degree*of*the kernel*of*any k-derivation (not necessarily locally*nilpotent*). ... Given*a*polynomial ®(.x) over the valuation*ring**of**a*local field k, the*algorithm*returns all irreducible factors y(x)*of*®(x), together with an integral*basis*for the quotient*ring*k[x]/(y(x)). ...##
###
Classification of Ding's Schubert Varieties: Finer Rook Equivalence

2007
*
Canadian Journal of Mathematics - Journal Canadien de Mathematiques
*

The crux

doi:10.4153/cjm-2007-002-9
fatcat:rtvugcjzwzd5vd56wfxwfa32te
*of*our approach is studying the*nilpotence*orders*of*linear forms in the cohomology*ring*. ... We classify the varieties X\ up*to*isomorphism, distinguishing them by their graded cohomology*rings*with integer coefficients. ... Proof It is easy*to*see that the elements*of*(7) form*a*reduced*Grobner**basis*with respect*to*the lexicographic order for whatever ideal they*generate*. ...##
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Construction of Finitely Presented Lie Algebras and Superalgebras

1996
*
Journal of symbolic computation
*

The presentation

doi:10.1006/jsco.1996.0016
fatcat:gemyc4cdjzc4pdj3yn4y7zxbvu
*of*Lie (super)algebras by*a*finite set*of**generators*and defining relations is one*of*the most*general*mathematical and*algorithmic*schemes*of*their analysis. ... We describe here an*algorithm*for constructing the*basis**of**a*finitely presented Lie (super)algebra and its commutator table, and its implementation in C. ... Acknowledgments We are grateful*to*J. Backelin, P. Gragert, D. Leites, A.A. Mikhalev, V. Robuk, M. Roelofs, and especially*to*V. Ufnarovsky for fruitful discussions and useful remarks. ...
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