Related papers: The EGH Conjecture and the Sperner property of com…
We show that each perfect matching in a bipartite graph $G$ intersects at least half of the perfect matchings in $G$. This result has equivalent formulations in terms of the permanent of the adjacency matrix of a graph, and in terms of…
We classify all complete uniform multipartite hypergraphs with respect to some algebraic properties, such as being (almost) complete intersection, Gorenstein, level, $l$-Cohen-Macaulay, $l$-Buchsbaum, unmixed, and satisfying Serre's…
In this paper we study conditions which guarantee the existence of perfect matchings and perfect fractional matchings in uniform hypergraphs. We reduce this problem to an old conjecture by Erd\H{o}s on estimating the maximum number of edges…
We discuss the conjecture that every maximal Hardy field has the Intermediate Value Property for differential polynomials, and its equivalence to the statement that all maximal Hardy field are elementarily equivalent to the differential…
A homogeneous ideal $I$ of a polynomial ring $S$ is said to have the Rees property if, for any homogeneous ideal $J \subset S $ which contains $I$, the number of generators of $J$ is smaller than or equal to that of $I$. A homogeneous ideal…
We prove, in any positive characteristic, Parseval-Rayleigh identities for the residue map of a homogeneous complete intersection. As an application, we give a conceptual proof of the folklore fact that generic homogeneous complete…
We say that a graph $G$ has a perfect $H$-packing if there exists a set of vertex-disjoint copies of $H$ which cover all the vertices in $G$. The seminal Hajnal--Szemer\'edi theorem characterises the minimum degree that ensures a graph $G$…
Let $K$ be a field, $A$ a standard graded $K$-algebra and $M$ a finitely generated graded $A$-module. Inspired by our previous works, we study the Hilbert depth of $h_M$, that is $$\operatorname{hdepth}(h_M)=\max\{d\;:\; \sum\limits_{j\leq…
In his preprint ``Differential-Geometric Characterizations of Complete Intersections'' (alg-geom/9407002), J.M.Landsberg introduces an elementary characterization of complete intersections. The proof of this criterion uses the method of…
Let $G$ be a graph and $\mathcal{H}$ be a hypergraph both on the same vertex set. We say that a hypergraph $\mathcal{H}$ is a \emph{Berge}-$G$ if there is a bijection $f : E(G) \rightarrow E(\mathcal{H})$ such that for $e \in E(G)$ we have…
A graph $G$ has the \emph{Perfect Matching Hamiltonian property} (or for short, $G$ is $PMH$) if, for each one of its perfect matchings, there is another perfect matching of $G$ such that the union of the two perfect matchings yields a…
We address the conjecture of [Durfee1978], bounding the singularity genus, p_g, by a multiple of the Milnor number, \mu, for an n-dimensional isolated complete intersection singularity. We show that the original conjecture of Durfee, namely…
When $I$ is the edge ideal of a graph $G$, we use combinatorial properities, particularly Property $P$ on connectivity of neighbors of an edge, to classify when a binomial sum of vertices is a regular element on $R/I(G)$. Under a mild…
We say that a (di)graph $G$ has a perfect $H$-packing if there exists a set of vertex-disjoint copies of $H$ which cover all the vertices in $G$. The seminal Hajnal--Szemer\'edi theorem characterises the minimum degree that ensures a graph…
Stanley proved that, in characteristic zero, all artinian monomial complete intersections have the strong Lefschetz property. We provide a positive characteristic complement to Stanley's result in the case of artinian monomial complete…
In this article we introduce and study the intersection graph of graded ideals of graded rings. The intersection graph of $G-$graded ideals of a graded ring $(R,G)$ is a simple graph, denoted by $Gr_G(R)$, whose vertices are the nontrivial…
We find an algorithm to compute the quadratic Euler characteristic of a smooth projective complete intersection of hypersurfaces of the same degree. As an example, we compute the quadratic Euler characteristic of a smooth projective…
It has been conjectured by Eisenbud, Green and Harris that if $I$ is a homogeneous ideal in $k[x_1,...,x_n]$ containing a regular sequence $f_1,...,f_n$ of degrees $\deg(f_i)=a_i$, where $2\leq a_1\leq ... \leq a_n$, then there is a…
Using class field theory, we prove a restriction on the intersection of the maximal abelian extensions associated with different number fields. This restriction is then used to improve a result of Rosen and Silverman about the linear…
In this paper, we provide a combinatorial characterization of those collections of cells whose inner $2$-minor ideals are complete intersections. More precisely, given a collection of cells $\mathcal C$ and its associated inner $2$-minor…