A lattice formulation of the F4 completion procedure

We write a procedure for constructing noncommutative Groebner bases. Reductions are done by particular linear projectors, called reduction operators. The operators enable us to use a lattice construction to reduce simultaneously each S-polynomial into a unique normal form. We write an implementation as well as an example to illustrate our procedure. Moreover, the lattice construction is done by Gaussian elimination, which relates our procedure to the F4 algorithm for constructing commutative Groebner bases

Regularity and K0-group of quadric solvable polynomial algebras

Cataloged from PDF version of article.Concerning solvable polynomial algebras in the sense of Kandri-Rody and Weispfenning [J. Symbolic Comput. 9 (1990) 1–26], it is shown how to recognize and construct quadric solvable polynomial algebras in an algorithmic way. If A=k[a1,…,an] is a quadric solvable polynomial algebra, it is proved that gl.dimA⩽n and Full-size image (<1 K). If A is a tame quadric solvable polynomial algebra, it is shown that A is completely constructable and Auslander regular

CHAMP: A Cherednik Algebra Magma Package

We present a computer algebra package based on Magma for performing computations in rational Cherednik algebras at arbitrary parameters and in Verma modules for restricted rational Cherednik algebras. Part of this package is a new general Las Vegas algorithm for computing the head and the constituents of a module with simple head in characteristic zero which we develop here theoretically. This algorithm is very successful when applied to Verma modules for restricted rational Cherednik algebras and it allows us to answer several questions posed by Gordon in some specific cases. We could determine the decomposition matrices of the Verma modules, the graded G-module structure of the simple modules, and the Calogero-Moser families of the generic restricted rational Cherednik algebra for around half of the exceptional complex reflection groups. In this way we could also confirm Martino's conjecture for several exceptional complex reflection groups.Comment: Final version to appear in LMS J. Comput. Math. 41 pages, 3 ancillary files. CHAMP is available at http://thielul.github.io/CHAMP/. All results are listed explicitly in the ancillary PDF document (currently 935 pages). Please check the website for further update

Signature Gr\"obner bases in free algebras over rings

We generalize signature Gr\"obner bases, previously studied in the free algebra over a field or polynomial rings over a ring, to ideals in the mixed algebra $R[x_1,...,x_k]\langle y_1,\dots,y_n \rangle$ where $R$ is a principal ideal domain. We give an algorithm for computing them, combining elements from the theory of commutative and noncommutative (signature) Gr\"obner bases, and prove its correctness. Applications include extensions of the free algebra with commutative variables, e.g., for homogenization purposes or for performing ideal theoretic operations such as intersections, and computations over $\mathbb{Z}$ as universal proofs over fields of arbitrary characteristic. By extending the signature cover criterion to our setting, our algorithm also lifts some technical restrictions from previous noncommutative signature-based algorithms, now allowing, e.g., elimination orderings. We provide a prototype implementation for the case when $R$ is a field, and show that our algorithm for the mixed algebra is more efficient than classical approaches using existing algorithms.Comment: 10 page

An Axiomatic Setup for Algorithmic Homological Algebra and an Alternative Approach to Localization

In this paper we develop an axiomatic setup for algorithmic homological algebra of Abelian categories. This is done by exhibiting all existential quantifiers entering the definition of an Abelian category, which for the sake of computability need to be turned into constructive ones. We do this explicitly for the often-studied example Abelian category of finitely presented modules over a so-called computable ring $R$, i.e., a ring with an explicit algorithm to solve one-sided (in)homogeneous linear systems over $R$. For a finitely generated maximal ideal $\mathfrak{m}$ in a commutative ring $R$ we show how solving (in)homogeneous linear systems over $R_{\mathfrak{m}}$ can be reduced to solving associated systems over $R$. Hence, the computability of $R$ implies that of $R_{\mathfrak{m}}$. As a corollary we obtain the computability of the category of finitely presented $R_{\mathfrak{m}}$-modules as an Abelian category, without the need of a Mora-like algorithm. The reduction also yields, as a by-product, a complexity estimation for the ideal membership problem over local polynomial rings. Finally, in the case of localized polynomial rings we demonstrate the computational advantage of our homologically motivated alternative approach in comparison to an existing implementation of Mora's algorithm.Comment: Fixed a typo in the proof of Lemma 4.3 spotted by Sebastian Posu

How to obtain lattices from (f,σ,δ)-codes via a generalization of Construction A

We show how cyclic (f,σ,δ)-codes over finite rings canonically induce a Z-lattice in RN by using certain quotients of orders in nonassociative division algebras defined using the skew polynomial f. This construction generalizes the one using certain σ-constacyclic codes by Ducoat and Oggier, which used quotients of orders in non-commutative associative division algebras defined by f, and can be viewed as a generalization of the classical Construction A for lattices from linear codes. It has the potential to be applied to coset coding, in particular to wire-tap coding. Previous results by Ducoat and Oggier are obtained as special cases

Veronese and Segre morphisms between non-commutative projective spaces

Number theory, Algebra and Geometr