We find a minimal generating set for the De Concini-Procesi ideals indexed by hooks, and study th... more We find a minimal generating set for the De Concini-Procesi ideals indexed by hooks, and study their minimal free resolutions as well as their Hilbert series and regularity.
We use the definition of a simplicial cycle to define an odd-cycle-free facet complex (hypergraph... more We use the definition of a simplicial cycle to define an odd-cycle-free facet complex (hypergraph). These are facet complexes that do not contain any cycles of odd length. We show that besides one class of such facet complexes, all of them satisfy the König property. This new family of complexes includes the family of balanced hypergraphs, which are known to satisfy the König property. These facet complexes are, however, not Mengerian; we give an example to demonstrate this fact.
We generalize the concept of a cycle from graphs to simplicial complexes. We show that a simplici... more We generalize the concept of a cycle from graphs to simplicial complexes. We show that a simplicial cycle is either a sequence of facets connected in the shape of a circle, or is a cone over such a structure. We show that a simplicial tree is a connected cycle-free simplicial complex, and use this characterization to produce an algorithm that checks in polynomial time whether a simplicial complex is a tree. We also present an efficient algorithm for checking whether a simplicial complex is grafted, and therefore
We find a minimal generating set for the defining ideal of the schematic intersection of the set ... more We find a minimal generating set for the defining ideal of the schematic intersection of the set of diagonal matrices with the closure of the conjugacy class of a nilpotent matrix indexed by a hook partition. The structure of this ideal allows us to compute its minimal free resolution and give an explicit description of the graded Betti numbers, and study its Hilbert series and regularity.
Given a simple graph G on n vertices, we prove that it is possible to reconstruct several algebra... more Given a simple graph G on n vertices, we prove that it is possible to reconstruct several algebraic properties of the edge ideal from the deck of G, that is, from the collection of subgraphs obtained by removing a vertex from G. These properties include the Krull dimension, the Hilbert function, and all the graded Betti numbers i,j where j < n. We also state many further questions that arise from our study.
We find a minimal generating set for the De Concini-Procesi ideals indexed by hooks, and study th... more We find a minimal generating set for the De Concini-Procesi ideals indexed by hooks, and study their minimal free resolutions as well as their Hilbert series and regularity.
We use the definition of a simplicial cycle to define an odd-cycle-free facet complex (hypergraph... more We use the definition of a simplicial cycle to define an odd-cycle-free facet complex (hypergraph). These are facet complexes that do not contain any cycles of odd length. We show that besides one class of such facet complexes, all of them satisfy the König property. This new family of complexes includes the family of balanced hypergraphs, which are known to satisfy the König property. These facet complexes are, however, not Mengerian; we give an example to demonstrate this fact.
We generalize the concept of a cycle from graphs to simplicial complexes. We show that a simplici... more We generalize the concept of a cycle from graphs to simplicial complexes. We show that a simplicial cycle is either a sequence of facets connected in the shape of a circle, or is a cone over such a structure. We show that a simplicial tree is a connected cycle-free simplicial complex, and use this characterization to produce an algorithm that checks in polynomial time whether a simplicial complex is a tree. We also present an efficient algorithm for checking whether a simplicial complex is grafted, and therefore
We find a minimal generating set for the defining ideal of the schematic intersection of the set ... more We find a minimal generating set for the defining ideal of the schematic intersection of the set of diagonal matrices with the closure of the conjugacy class of a nilpotent matrix indexed by a hook partition. The structure of this ideal allows us to compute its minimal free resolution and give an explicit description of the graded Betti numbers, and study its Hilbert series and regularity.
Given a simple graph G on n vertices, we prove that it is possible to reconstruct several algebra... more Given a simple graph G on n vertices, we prove that it is possible to reconstruct several algebraic properties of the edge ideal from the deck of G, that is, from the collection of subgraphs obtained by removing a vertex from G. These properties include the Krull dimension, the Hilbert function, and all the graded Betti numbers i,j where j < n. We also state many further questions that arise from our study.
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Papers by Sara Faridi