The core of an ideal is the intersection of all its reductions. We describe the core of a zero-dimensional monomial ideal I as the largest monomial ideal contained in a general reduction of I. This provides a new interpretation of the core in the monomial case as well as an efficient algorithm for computing it. We relate the core to adjoints and first coefficient ideals, and in dimension two an...

Monomial representations and operations for Grr obner bases computations are investigated from an implementation point of view. The technique of vectorized monomial operations is introduced and it is shown how it expedites computations of Grr obner bases. Furthermore, a rank-based monomial representation and comparison technique is examined and it is concluded that this technique does not yield...

The paper presents two algorithms for finding irreducible decomposition of monomial ideals. The first one is recursive, derived from staircase structures of monomial ideals. This algorithm has a good performance for highly non-generic monomial ideals. The second one is an incremental algorithm, which computes decompositions of ideals by adding one generator at a time. Our analysis shows that th...

In this paper we primarily study monomial ideals and their minimal free resolutions by studying their associated lcm-lattices. In particular, we formally define the notion of coordinatizing a finite atomic lattice P to produce a monomial ideal whose lcm-lattice is P , and we give a characterization of all such coordinatizations. We prove that all relations in the lattice L(n) of all finite atom...

In recent work, Clark and Tchernev introduced the notion of monomial resolution supported on a poset. In this talk, I will review this notion, and the related notion of monomial resolution supported on a CW complex. I will discuss a result of mine that shows that resolutions supported on a CW complex are also supported on the face poset of a CW complex. This relates to the work of Clark and Tch...

We characterize monomial ideals which are intersections of powers of monomial prime ideals and study classes of ideals with this property, among them polymatroidal ideals.

We study the Hochschild homology of algebras related via split pairs, and apply this to fibre products, trivial extensions, monomial algebras, graded-commutative algebras and quantum complete intersections. In particular, we compute lower bounds for the dimensions of both the Hochschild homology and cohomology groups of quantum complete intersections.

We show that the number of elements generating a squarefree monomial ideal up to radical can always be bounded above in terms of the number of its minimal monomial generators and the maximal height of its minimal primes.

We characterize the approximate monomial complexity, sign monomial complexity, and the approximate L1 norm of symmetric functions in terms of simple combinatorial measures of the functions. Our characterization of the approximate L1 norm solves the main conjecture in [AFH12]. As an application of the characterization of the sign monomial complexity, we prove a conjecture in [ZS09] and provide a...

This paper gives an overview of various forms of the differential equations of exterior, interior and terminal ballistics, as well as a presentation of the problems they induce in computer algebra. Some of these problems have been investigated at ISL (French-German Institute of Saint-Louis): through the quasi-monomial transforms (QMT) of Brenig, for the ordinary differential equations (ODE’s) o...