Mathematics

Let D be a weighted oriented graph and I(D) be its edge ideal. In this paper, we investigate the Betti numbers of I(D) via upper-Koszul simplicial complexes, Betti splittings and the mapping cone construction. In particular, we provide recursive formulas for the Betti numbers of edge ideals of several classes of weighted oriented graphs. We also identify classes of weighted oriented graphs whose edge ideals have a unique extremal Betti number which allows us to compute the regularity and projective dimension for the identified classes. Furthermore, we characterize the structure of a weighted oriented graph D on n vertices such that pdim (R/I(D))=n where R=k[ x 1 ,…, x n ] .

Let G be a graph on [n] and J G be the binomial edge ideal of G in the polynomial ring S=K[ x 1 ,?? x n , y 1 ,?? y n ] . In this paper we investigate some topological properties of a poset associated to the minimal primary decomposition of J G . We show that this poset admits some specific subposets which are contractible. This in turn, provides some interesting algebraic consequences. In particular, we characterize all graphs G for which depthS/ J G =4 .

A \emph{congruence} on N n is an equivalence relation on N n that is compatible with the additive structure. If k is a field, and I is a \emph{binomial ideal} in k[ X 1 ,…, X n ] (that is, an ideal generated by polynomials with at most two terms), then I induces a congruence on N n by declaring u and v to be equivalent if there is a linear combination with nonzero coefficients of X u and X v that belongs to I . While every congruence on N n arises this way, this is not a one-to-one correspondence, as many binomial ideals may induce the same congruence. Nevertheless, the link between a binomial ideal and its corresponding congruence is strong, and one may think of congruences as the underlying combinatorial structures of binomial ideals. In the current literature, the theories of binomial ideals and congruences on N n are developed separately. The aim of this survey paper is to provide a detailed parallel exposition, that provides algebraic intuition for the combinatorial analysis of congruences. For the elaboration of this survey paper, we followed mainly [Kahle and Miller, Algebra Number Theory 8(6):1297-1364, 2014] with an eye on [Eisenbud and Sturmfels. Duke Math J 84(1):1-45, 1996] and [Ojeda and Piedra Sánchez, J. Symbolic Comput 30(4):383-400, 2000].

We show that the fiber cones of general rational normal scrolls are Cohen--Macaulay and compute their Castelnuovo--Mumford regularities. Then we study the secant varieties of balanced rational normal scrolls. We describe the defining equations of their associated Rees algebras and compute the Castelnuovo--Mumford regularities of their fiber cones.

In this article, we obtain an upper bound for the regularity of the binomial edge ideal of a graph whose every block is either a cycle or a clique. As a consequence, we obtain an upper bound for the regularity of binomial edge ideal of a cactus graph. We also identify certain subclass attaining the upper bound.

We make explicit the exponential bound on the degrees of the polynomials appearing in the Effective Quillen-Suslin Theorem, and apply it jointly with the Hilbert-Burch Theorem to show that the syzygy module of a sequence of m polynomials in n variables defining a complete intersection ideal of grade two is free, and that a basis of it can be computed with bounded degrees. In the known cases, these bounds improve previous results.

Let I be a homogeneous ideal in a polynomial ring S . In this paper, we extend the study of the asymptotic behavior of the minimum distance function δ I of I and give bounds for its stabilization point, r I , when I is an F -pure or a square-free monomial ideal. These bounds are related with the dimension and the Castelnuovo--Mumford regularity of I .

We consider the following question: if a simplicial complex Δ has d -homology, then does the corresponding d -cycle always induce cycles of smaller dimension that are not boundaries in Δ ? We provide an answer to this question in a fixed dimension. We use the breaking of homology to show the subadditivity property for the maximal degrees of syzygies of monomial ideals in a fixed homological degree.

Let (A,m) be a Noetherian local ring of dimension d>0 and I an I -primary ideal of A . In this paper, we discuss a sufficient condition, for the Buchsbaumness of the local ring A to be passed onto the associated graded ring of filtration. Let I denote an I -good filtration. We prove that if A is Buchsbaum and the I-invariant, I(A) and I(G(I)) , coincide then the associated graded ring G(I) is Buchsbaum. As an application of our result, we indicate an alternative proof of a conjecture, of Corso on certain boundary conditions for Hilbert coefficients.

The article proved the upper bound of leading coefficient of characteristic polynomial of graded ideal in a ring of generalized polynomials.