Syzygies among reduction operators

We introduce the notion of syzygy for a set of reduction operators and relate it to the notion of syzygy for presentations of algebras. We give a method for constructing a linear basis of the space of syzygies for a set of reduction operators. We interpret these syzygies in terms of the confluence property from rewriting theory. This enables us to optimise the completion procedure for reduction operators based on a criterion for detecting useless reductions. We illustrate this criterion with an example of construction of commutative Gr{\"o}bner basis

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

05021 Abstracts Collection -- Mathematics, Algorithms, Proofs

From 09.01.05 to 14.01.05, the Dagstuhl Seminar 05021 ``Mathematics, Algorithms, Proofs\u27\u27 was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. LinkstFo extended abstracts or full papers are provided, if available

Twisted cohomology and likelihood ideals

A likelihood function on a smooth very affine variety gives rise to a twisted de Rham complex. We show how its top cohomology vector space degenerates to the coordinate ring of the critical points defined by the likelihood equations. We obtain a basis for cohomology from a basis of this coordinate ring. We investigate the dual picture, where twisted cycles correspond to critical points. We show how to expand a twisted cocycle in terms of a basis, and apply our methods to Feynman integrals from physics.Comment: 28 pages, 2 figures, comments are welcom

Explicit formula for the generating series of diagonal 3D rook paths

Let $a_n$ denote the number of ways in which a chess rook can move from a corner cell to the opposite corner cell of an $n \times n \times n$ three-dimensional chessboard, assuming that the piece moves closer to the goal cell at each step. We describe the computer-driven \emph{discovery and proof} of the fact that the generating series $G(x)= \sum_{n \geq 0} a_n x^n$ admits the following explicit expression in terms of a Gaussian hypergeometric function: G(x) = 1 + 6 \cdot \int_0^x \frac{\,\pFq21{1/3}{2/3}{2} {\frac{27 w(2-3w)}{(1-4w)^3}}}{(1-4w)(1-64w)} \, dw.Comment: To appear in "S\'eminaire Lotharingien de Combinatoire

A module minimization approach to Gabidulin decoding via interpolation

We focus on iterative interpolation-based decoding of Gabidulin codes and present an algorithm that computes a minimal basis for an interpolation module. We extend existing results for Reed-Solomon codes in showing that this minimal basis gives rise to a parametrization of elements in the module that lead to all Gabidulin decoding solutions that are at a fixed distance from the received word. Our module-theoretic approach strengthens the link between Gabidulin decoding and Reed-Solomon decoding, thus providing a basis for further work into Gabidulin list decoding

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

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