Minimal polynomial systems for parametric matrices

In this article, we study the minimal polynomials of parametric matrices. Using the concept of (comprehensive) Gröbner systems for parametric ideals, we introduce the notion of a minimal polynomial system for a parametric matrix, i.e. we decompose the space of parameters into a finite set of cells and for each cell we give the corresponding minimal polynomial of the matrix. We also present an algorithm for computing a minimal polynomial system for a given parametric matrix.     

Gröbner Bases for the Polynomial Ring with Infinite Variables and Their Applications

We develop the theory of Gröbner bases for ideals in a polynomial ring with countably infinite variables over a field. As an application we reconstruct some of the one-to-one correspondences among various sets of partitions by using the division algorithm.     

Universal Associative Envelopes of Nonassociative Triple Systems

We construct universal associative envelopes for the nonassociative triple systems arising from the trilinear operations of Bremner and Peresi applied to the 2-dimensional simple associative triple system. We use noncommutative Gröbner bases to determine monomial bases, structure constants, and centers of the universal envelopes. We show that the infinite dimensional envelopes are closely related to the down-up algebras of Benkart and Roby. For the finite dimensional envelopes, we determine the Wedderburn decompositions and classify the irreducible representations.     

NON-COMMUTATIVE GRÖBNER BASES UNDER COMPOSITION

Polynomial composition is the operation of replacing the variables in a polynomial with other polynomials. In this paper we give sufficient and necessary conditions on a set Θ of non-commutative polynomials to assure that the set G ○ Θ of composed polynomials is a Gröbner basis in the free associative algebra whenever G is. The subject was initiated by Hong, treating the commutative analogue in (1998, J. Symb. Comput. 25, 643–663).     

Gröbner Bases for Ideals of σ-PBW Extensions

In this article, we introduce the σ-PWB extensions and construct the theory of Gröbner bases for the left ideals of them. We prove the Hilbert's basis theorem and the division algorithm for this more general class of Poincaré–Birkhoff–Witt extensions. For the particular case of bijective and quasi-commutative σ-PWB extensions, we implement the Buchberger's algorithm for computing Gröbner bases of left ideals.     

Symbolic Differential Elimination for Symmetry Analysis

Differential problems are ubiquitous in mathematical modeling of physical and scientific problems. Algebraic analysis of differential systems can help in determining qualitative and quantitative properties of solutions of such systems. In this tutorial paper we describe several algebraic methods for investigating differential systems.     

Universal Associative Envelopes of (n+ 1)-Dimensional n-Lie Algebras

For n even, we prove Pozhidaev's conjecture on the existence of associative enveloping algebras for simple n-Lie (Filippov) algebras. More generally, for n even and any (n + 1)-dimensional n-Lie algebra L, we construct a universal associative enveloping algebra U(L) and show that the natural map L → U(L) is injective. We use noncommutative Gröbner bases to present U(L) as a quotient of the free associative algebra on a basis of L and to obtain a monomial basis of U(L). In the last section, we provide computational evidence that the construction of U(L) is much more difficult for n odd.     

Amortized multi-point evaluation of multivariate polynomials

The evaluation of a polynomial at several points is called the problem of multi-point evaluation. Sometimes, the set of evaluation points is fixed and several polynomials need to be evaluated at this set of points. Several efficient algorithms for this kind of “amortized” multi-point evaluation have been developed recently for the special cases of bivariate polynomials or when the set of evaluation points is generic. In this paper, we extend these results to the evaluation of polynomials in an arbitrary number of variables at an arbitrary set of points. We prove a softly linear complexity bound when the number of variables is fixed. Our method relies on a novel quasi-reduction algorithm for multivariate polynomials, that operates simultaneously with respect to several orderings on the monomials.     

Ideals Generated by Diagonal 2-Minors

With a simple graph G on [n], we associate a binomial ideal P_G generated by diagonal minors of an n × n matrix X = (x_ij) of variables. We show that for any graph G, P_G is a prime complete intersection ideal and determine the divisor class group of K[X]/P_G. By using these ideals, one may find a normal domain with free divisor class group of any given rank.     

Gröbner–Shirshov Bases for Schreier Extensions of Groups

In this article, by using the Gröbner–Shirshov bases, we give characterizations of the Schreier extensions of groups when the group is presented by generators and relations. An algorithm to find the conditions of a group to be a Schreier extension is obtained. By introducing a special total order, we obtain the structure of the Schreier extension by an HNN group.     

Full Graphs and Koszul Algebras I

The main result of this article is the establishment of a new connection between combinatorics and noncommutative algebra. This is done by linking a certain class of directed graphs, called full graphs, to quotients of path algebras that are Koszul algebras.     

On the peirce decompositions for freudenthal—kantor triple systems

It is the object of this paper to investigate the Peirce decomposition for Freudenthal-Kantor triple systems, by making use of the concept of anti-derivation of the triple systems.     

Fragments with finiteness conditions in particular over group rings

Abstract length functions on groups were introduced by Lyndon [5]. In [2] Chiswell showed that for a group action on an IR-tree there is a length function associated with each point of the tree, and moreover any length function can occur in this way. This paper establishes necessary and sufficient conditions for two length functions to arise from the same action of a group on a tree.