Looking for Groebner Basis Theory for (Almost) Skew 2-Nomial Algebras

In this paper, we introduce (almost) skew 2-nomial algebras and look for a one-sided or two-sided Gr\"obner basis theory for such algebras at a modest level. That is, we establish the existence of a skew multiplicative $K$-basis for every skew 2-nomial algebra, and we explore the existence of a (left, right, or two-sided) monomial ordering for an (almost) skew 2-nomial algebra. As distinct from commonly recognized algebras holding a Gr\"obner basis theory (such as algebras of the solvable type [K-RW] and some of their homomorphic images), a subclass of skew 2-nomial algebras that have a left Gr\"obner basis theory but may not necessarily have a two-sided Gr\"obner basis theory, respectively a subclass of skew 2-nomial algebras that have a right Gr\"obner basis theory but may not necessarily have a two-sided Gr\"obner basis theory, are determined such that numerous quantum binomial algebras (which provide binomial solutions to the Yang-baxter equation [Laf], [G-I2]) are involved.Comment: Revised version, 35 page

Buchberger-Zacharias Theory of multivariate Ore extensions

open2Following the recent survey on Buchberger-Zacharias Theory for monoid rings R[S] over a unitary effective ring R and an effective monoid S, we propose here a presentation of Buchberger-Zacharias Theory and related Gröbner basis computation algorithms for multivariate Ore extensions of rings presented as modules over a principal ideal domain, using Möller-Pritchard lifting theorem.openCeria, Michela; Mora, TeoCeria, Michela; Mora, Ferdinand

Buchberger-Zacharias Theory of multivariate Ore extensions

Following the recent survey on Buchberger-Zacharias Theory for monoid rings R[S] over a unitary effective ring R and an effective monoid S, we propose here a presentation of Buchberger Zacharias Theory and related Grobner basis computation algorithms for multivariate Ore extensions of rings presented as modules over a principal ideal domain, using Moller-Pritchard lifting theorem

Computational Ideal Theory in Finitely Generated Extension Rings

One of the most general extensions of Buchberger's theory of Gröbner bases is the concept of graded structures due to Robbiano and Mora. But in order to obtain algorithmic solutions for the computation of Göbner bases it needs additional computability assumptions. In this paper we introduce natural graded structures of finitely generated extension rings and present subclasses of such structures which allow uniform algorithmic solutions of the basic problems in the associated graded ring and, hence, of the computation of Gröbner bases with respect to the graded structure. Among the considered rings there are many of the known generalizations. But, in addition, a wide class of rings appears first time in the context of algorithmic Gröbner basis computations. Finally, we discuss which conditions could be changed in order to find further effective Gröbner structures and it will turn out that the most interesting constructive instances of graded structures are covered by our results

Computational ideal theory in finitely generated extension rings

One of the most general extensions of Buchberger's theory of Gröbner bases is the concept of graded structures due to Robbiano and Mora. But in order to obtain algorithmic solutions for the computation of Gröbner bases it needs additional computability assumptions. In this paper we introduce natural graded structures of finitely generated extension rings and present subclasses of such structures which allow uniform algorithmic solutions of the basic problems in the associated graded ring and, hence, of the computation of Gröbner bases with respect to the graded structure. Among the considered rings there are many of the known generalizations. But, in addition, a wide class of rings appears first time in the context of algorithmic Gröbner basis computations. Finally, we discuss which conditions could be changed in order to find further effective Gröbner structures and it will turn out that the most interesting constructive instances of graded structures are covered by our results

Computational Ideal Theory in Finitely Generated Extension Rings

One of the most general extensions of Buchberger's theory of Gröbner bases is the concept of graded structures due to Robbiano and Mora. But in order to obtain algorithmic solutions for the computation of Göbner bases it needs additional computability assumptions. In this paper we introduce natural graded structures of finitely generated extension rings and present subclasses of such structures which allow uniform algorithmic solutions of the basic problems in the associated graded ring and, hence, of the computation of Gröbner bases with respect to the graded structure. Among the considered rings there are many of the known generalizations. But, in addition, a wide class of rings appears first time in the context of algorithmic Gröbner basis computations. Finally, we discuss which conditions could be changed in order to find further effective Gröbner structures and it will turn out that the most interesting constructive instances of graded structures are covered by our results