Related papers: Implementing Gr\"obner bases for operads
A Gr\"obner basis for the ideal determining mod 2 cohomology of Grassmannian G_{3,n} is obtained. This is used, along with the method of obstruction theory, to establish some new immersion results for these manifolds.
In this paper we present the formal, computer-supported verification of a functional implementation of Buchberger's critical-pair/completion algorithm for computing Gr\"obner bases in reduction rings. We describe how the algorithm can be…
This is an expository article about operads in homotopy theory written as a chapter for an upcoming book. It concentrates on what the author views as the basic topics in the homotopy theory of operadic algebras: the definition of operads,…
Quite recently, Bremner et al. introduced a new approach to Rota's Classification Problem and classified some (new) operated polynomial identities. In this paper, we prove that all operated polynomial identities classified by Bremner et al.…
Recent results on the construction and applications of the transmutation (transformation) operators are discussed. Three new representations for solutions of the one-dimensional Schr\"odinger equation are considered. Due to the fact that…
With this paper we present an extension of our recent ISSAC paper about computations of Groebner(-Shirshov) bases over free associative algebras Z<X>. We present all the needed proofs in details, add a part on the direct treatment of the…
We develop a Gr\"obner basis theory for a class of algebras that generalizes both PBW-algebras and rings of differential algebras on smooth varieties. Emphasis lies on methods to compute filtrations and graded structures defined by weight…
Let $\Lambda$ be a commutative Noetherian ring, and let $I$ be a proper ideal of $\Lambda$, $R=\Lambda /I$. Consider the polynomial rings $T=\Lambda [x_1,...x_n]$ and $A=R[x_1,...,x_n]$. Suppose that linear equations are solvable in…
This paper involves categories and computer science. The paper is motivated by a question which arises from two pieces of research. Firstly, the work of Brown and Heyworth which extends rewriting techniques to enable the computation of left…
Grassmann manifolds $G_{k,n}$ are among the central objects in geometry and topology. The Borel picture of the mod 2 cohomology of $G_{k,n}$ is given as a polynomial algebra modulo a certain ideal $I_{k,n}$. The purpose of this paper is to…
In this paper, we characterized the relationship between Groebner bases and u-bases: any minimal Groebner basis of the syzygy module for n univariate polynomials with respect to the term-over-position monomial order is its u-basis.…
Solving zero-dimensional polynomial systems using Gr\"obner bases is usually done by, first, computing a Gr\"obner basis for the degree reverse lexicographic order, and next computing the lexicographic Gr\"obner basis with a change of order…
We give an introduction to the theory of initial ideals and initial algebras with emphasis on the transfer of structural properties.
Signature-based algorithms are the latest and most efficient approach as of today to compute Gr\"obner bases for polynomial systems over fields. Recently, possible extensions of these techniques to general rings have attracted the attention…
In this work, it is proposed a method for computing Noncommutative Gr\"obner bases over a valuation n{\oe}therian ring. We have generalized the fundamental theorem on normal forms over an arbitrary ring. The classical method of dynamical…
There are several efficient methods to solve linear interval polynomial systems in the context of interval computations, however, the general case of interval polynomial systems is not yet covered as well. In this paper we introduce a new…
Gr\"obner bases, in their noncommutative version, and word reversing are methods for solving the word problem of a presented monoid, and both rely on iteratively completing the initial list of relations. Simple examples may suggest to…
In this paper we present a right version of the algorithms developed for to compute Gr\"obner bases over bijective skew PBW extensions in the left case given in [3]. In particular, we adapt the theory of reduction and we build a right…
The Gr\"obner basis detection (GBD) is defined as follows: Given a set of polynomials, decide whether there exists -and if "yes" find- a term order such that the set of polynomials is a Gr\"obner basis. This problem was shown to be NP-hard…
Signature-based algorithms have become a standard approach for Gr\"obner basis computations for polynomial systems over fields, but how to extend these techniques to coefficients in general rings is not yet as well understood. In this…