In [1] it is shown that every monomial ideal admits a simplicial resolution (Taylor’s resolution) and that some minimal free resolutions are supported in simplicial complexes (Scarf ideals, monomial regular sequences). This idea is generalized in [2] where cellular resolutions are introduced. The authors show that every monomial ideal admits a resolution supported in a regular cell complex (the...

The Hilbert series of local cohomologies for monomial ideals, which are not necessarily square-free, is established. As applications, we give a sharp lower bound of the non-vanishing degree of local cohomologies and also a sharp lower bound of the positive integer k of k-Buchsbaumness for generalized CohenMacaulay monomial ideals.

We give equivalent conditions for a monomial sequence to be a dsequence or a proper sequence, and a sufficient condition for a monomial sequence to be an s-sequence in order to compute invariants of the symmetric algebra of the ideal generated by it.

This article studies jet schemes of monomial schemes. They are known to be equidimensional but usually are not reduced [13]. We thus investigate their structure further, giving a formula for the multiplicity along every component of the jet schemes of a general reduced monomial hypersurface (that is, the case of a simple normal crossing divisor).

We classify all monomial planar Cremona maps by multidegree using recent methods developed by Aluffi. Following the main result, we prove several more properties of the set of these maps, and also extend the results to the more general ‘r.c. monomial’ maps.

We study Stanley decompositions and show that Stanley’s conjecture on Stanley decompositions implies his conjecture on partitionable Cohen-Macaulay simplicial complexes. We also prove these conjectures for all Cohen-Macaulay monomial ideals of codimension 2 and all Gorenstein monomial ideals of codimension 3.

We show that the defining ideal of every monomial curve in the affine or projective three-dimensional space can be set-theoretically defined by three binomial equations, two of which set-theoretically define a determinantal ideal generated by the 2-minors of a 2× 3 matrix with monomial entries.

Irreducible decomposition of monomial ideals has an increasing number of applications from biology to pure math. This paper presents the Slice Algorithm for computing irreducible decompositions, Alexander duals and socles of monomial ideals. The paper includes experiments showing good performance in practice.

In this work, we consider an interesting variant of the well studied KP model for selfish routing on parallel links, which reflects some influence from the much older Wardrop model [29]. In the new model, user traffics are still unsplittable and links are identical. Social Cost is now the expectation of the sum, over all links, of Latency Costs; each Latency Cost is modeled as a certain polynom...

We present in full details a version of the DinurShamir Cube Attack (Dinur & Shamir 2009) for a generic finite field of order q. In particular, when applied to multivariate monomials of degree d in k < d variables, the attack acts exactly in the same way if the selected monomial was using the degree k monomial in the same k variables.