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

Rafał Abłamowicz
2010 Advances in Applied Clifford Algebras  
Mora who already in the 1990s considered 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.  ... 
doi:10.1007/s00006-010-0205-0 fatcat:a3ronds35vf6vifhyuks5t6c5u

Index to Volume 41

2006 Journal of symbolic computation  
devoted 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  ... 
doi:10.1016/s0747-7171(06)00119-2 fatcat:n3voc3w2dvex7bo3gopguifv4m

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

Samiha Monceur, Ihsen Yengui
2012 Journal of Algebra  
Let R be 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.  ... 
doi:10.1016/j.jalgebra.2011.11.015 fatcat:kvpxyomqkngdraa6776c6ymahy

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

Klaus Madlener, Birgit Reinert
1998 Journal of symbolic computation  
Using this reduction we present 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  ... 
doi:10.1006/jsco.1997.0165 fatcat:qw5nz5hlqbhylpbsbkfzriv2uy

Characterization of rank two locally nilpotent derivations in dimension three [article]

Moulay A. Barkatou, Hassan El Houari, M'hammed El Kahoui
2007 arXiv   pre-print
Together with an 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 ].  ... 
arXiv:0705.2489v1 fatcat:5fgutkj46rgp7b7v4rwwxaju3i

The isomorphism problem for graded algebras and its application to mod-p cohomology rings of small p-groups [article]

Bettina Eick, Simon King
2015 arXiv   pre-print
The mod-p cohomology 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.  ... 
arXiv:1503.04666v1 fatcat:owfi4grsq5ajxneztc2fhkflwm

A Divide and Conquer Method to Compute Binomial Ideals [chapter]

Deepanjan Kesh, Shashank K. Mehta
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] .  ... 
doi:10.1007/978-3-642-54423-1_56 fatcat:wframq5albavfnqtowl6m7usny

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

Eimear Byrne, Teo Mora
2009 Gröbner Bases, Coding, and Cryptography  
We discuss and compare the main 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]. •  ... 
doi:10.1007/978-3-540-93806-4_14 fatcat:5o2prs35uzfmtf4ghdp5cfumwu

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]

Gregor Kemper
2014 arXiv   pre-print
The main purpose 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.  ... 
arXiv:1310.6851v2 fatcat:infhszd6lnh2ll5zc73lbgmmda

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

Mike Develin, Jeremy L. Martin, Victor Reiner
2007 Canadian Journal of Mathematics - Journal Canadien de Mathematiques  
The crux 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.  ... 
doi:10.4153/cjm-2007-002-9 fatcat:rtvugcjzwzd5vd56wfxwfa32te

Construction of Finitely Presented Lie Algebras and Superalgebras

VLADIMIR P. GERDT, VLADIMIR V KORNYAK
1996 Journal of symbolic computation  
The presentation 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.  ... 
doi:10.1006/jsco.1996.0016 fatcat:gemyc4cdjzc4pdj3yn4y7zxbvu
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