Skew polynomial rings, Gröbner bases and the letterplace embedding of the free associative algebra

Abstract

In this paper we introduce an algebra embedding ι : K X → S from the free associative algebra K X generated by a finite or countable set X into the skew monoid ring S = P * Σ defined by the commutative polynomial ring P = K[X × N * ] and by the monoid Σ = σ generated by a suitable endomorphism σ : P → P . If P = K[X] is any ring of polynomials in a countable set of commuting variables, we present also a general Gröbner bases theory for graded two-sided ideals of the graded algebra S = i S i with S i = P σ i and σ : P → P an abstract endomorphism satisfying compatibility conditions with ordering and divisibility of the monomials of P . Moreover, using a suitable grading for the algebra P compatible with the action of Σ, we obtain a bijective correspondence, preserving Gröbner bases, between graded Σ-invariant ideals of P and a class of graded two-sided ideals of S. By means of the embedding ι this results in the unification, in the graded case, of the Gröbner bases theories for commutative and non-commutative polynomial rings. Finally, since the ring of ordinary difference polynomials P = K[X × N] fits the proposed theory one obtains that, with respect to a suitable grading, the Gröbner bases of finitely generated graded ordinary difference ideals can be computed also in the operators ring S and in a finite number of steps up to some fixed degree.2000 Mathematics Subject Classification. Primary 16Z05. Secondary 13P10, 68W30.

Multigraded Hilbert series of noncommutative modules

“…Note that internally to the kernel, the implementation uses indeed commutative analogues of noncommutative monomial ideals according to the "letterplace correspondence". For more details about the wide scope of letterplace methods we refer to [15,16,17,19].…”

“…To show the performance of our implementation, we have carried out the computations on a Dell PowerEdge R720 with two Intel(R) Xeon(R) CPU E5-2690 @ 2.90GHz, 20 MB Cache, 16 Cores, 32 Threads, 192 GB RAM with a Linux operating system (Gentoo). Besides the experimental implementation in [18], this is the first implementation in the kernel of a computer algebra system performing the computation of Hilbert series of noncommutative algebras in general.…”

Computing minimal free resolutions of right modules over noncommutative algebras

“…In case the resolution is infinite with respect to the Betti numbers or its length, the proposed method behaves algorithmically by bounding the degree of the syzygies to be computed. The idea to use commutative modules has its roots in a series of papers [7,8,9] where commutative analogues of noncommutative graded ideals has been introduced for the purpose of obtaining noncommutative Gröbner bases computations from commutative ones. In those papers such correspondence between ideals is called the letterplace correspondence.…”

Enhanced computations of gröbner bases in free algebras as a new application of the letterplace paradigm

“…Nevertheless the naive implementation performed nicely [LL09,SL13]. In this article we report on the work done for optimizing and generalizing this algorithm.…”