In this paper we consider diierent involutive divisions and describe their implementation in Mathematica together with algorithms for the construction of involutive bases for monomial ideals. As a straightforward application, we consider computation of the Hilbert function and the Hilbert polynomial for a monomial ideal, or for a polynomial one represented by its Grr obner basis. This allows on...

We present a vertex operator algebra which is an extension of the level k vertex operator algebra for the ŝl2 conformal field theory. We construct monomial basis of its irreducible representations.

Let μ = (μ1 ≥ μ2 ≥ · · · ≥ μk > 0) be a partition of n. We shall identify μ with its Ferrers diagram (using the French notation). To each cell s of the Ferrers diagram, we associate its coordinates (i, j), where i is the height of s and j the position of s in its row. The pairs (i − 1, j − 1) occurring while s describes μ will be briefly referred to as the set of the biexponents of μ. Now let (...

There is a natural infinite graph whose vertices are the monomial ideals in a polynomial ring K[x1, . . . , xn]. The definition involves Gröbner bases or the action of the algebraic torus (K∗)n. We present algorithms for computing the (affine schemes representing) edges in this graph. We study the induced subgraphs on multigraded Hilbert schemes and on square-free monomial ideals. In the latter...

We investigate the affine canonical basis ([L2]) and the monomial basis constructed in [LXZ] in Lusztig’s geometric setting. We show that the transition matrix between the two bases is upper triangular with 1’s in the diagonal and coefficients in the upper diagonal entries in Z≥0[v, v ]. As a consequence, we show that part of the monomial basis elements give rise to resolutions of support varie...

We consider three modifications of our involutive algorithm for computing Janet bases. These modifications are related to degree compatible monomial orders and specify selection strategies for non-multiplicative prolongations. By using the standard data base of polynomial benchmarks for Gröbner bases software we compare the modifications and confront them with Magma that implements Faugère’s F4...

While the computation of Gröbner bases is known to be an expspace-complete problem, the generic behaviour of algorithms for their computation is much better. We study generic properties of Gröbner bases and analyse precisely the best algorithm currently known, F5. 1. Gröbner Bases Gröbner bases are a fundamental tool in computational algebra. They provide a multivariate generalization of Euclid...

Sparse elimination exploits the structure of a set of multivariate polynomials by measuring complexity in terms of Newton polytopes. We examine polynomial systems that generate 0-dimensional ideals: a generic monomial basis for the coordinate ring of such a system is de ned from a mixed subdivision. We o er a simple proof of this known fact and relate the computation of a monomial basis to the ...

In the study of double-loop computer networks, the diagrams known as L-shapes arise as a graphical representation of an optimal routing for every graph’s node. The description of these diagrams provides an efficient method for computing the diameter and the average minimum distance of the corresponding graphs. We extend these diagrams to multiloop computer networks. For each Cayley digraph with...

We study Bott-Samelson varieties for the group GLn(C), their toric degenerations and standard monomial type bases for their homogeneous coordinate rings. A 3-dimensional example is described in detail.