An extension of Buchberger's algorithm and calculationsin enveloping fields of lie algebras
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Algorithm 628: An algorithm for constructing canonical bases of polynomial ideals
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Groebner Bases for Non-Commutative Polynomial Rings
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Gröbner bases with coefficients in rings
Journal of Symbolic Computation
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Gröbner bases and syzygies on bimodules over PBW algebras
Journal of Symbolic Computation
Gröbner bases and logarithmic D-modules
Journal of Symbolic Computation
Looking for Gröbner basis theory for (almost) skew 2-nomial algebras
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ICMS'06 Proceedings of the Second international conference on Mathematical Software
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Mathematical and Computer Modelling: An International Journal
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ACM Communications in Computer Algebra
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We introduce a class of non-commutative polynomial rings over fields intermediate betweencommutative polynomial rings and general non-commutative polynomial rings. This class of solvable polynomial rings includes many rings arising naturally in mathematics and physics, such as iterated Ore extensions of fields and enveloping algebras of finite dimensional Lie algebras. We present algorithms that compute Grobner bases of one- and two-sided ideals in solvable polynomial rings. They extend Buchberger's algorithm (see Buchberger, 1985) in the commutative case and Apel and Lassner's algorithms (see Apel & Lassner, 1988) for one-sided ideals in enveloping algebras of Lie algebras, as well as the results on one-sided standard bases in Weyl algebras, sketched in Galligo (1985). We show that reduced one- and two-sided Grobner bases in solvable polynomial rings are unique, and we solve the word problem and the ideal membership problem for algebras of solvable type, in particular in Clifford algebras. Further applications include the computation of elimination ideals, computing in residue modules and the computation of generators for modules of syzygies.