Related papers: A Minimal Poset Resolution of Stable Ideals
In this paper we extend the well-known iterated mapping cone procedure to monomial ideals in strongly Koszul algebras. We study properties of ideals generated by monomials in commutative Koszul algebras and show that the linear strand of…
Polynomial meshes (called sometimes "norming sets") allow us to estimate the supremum norm of polynomials on a fixed compact set by the norm on its discrete subset. We give a general construction of polynomial weakly admissible meshes on…
Squarefree monomial ideals arising from finite meet-semilattices and their free resolutions are studied. For the squarefree monomial ideals corresponding to poset ideals in a distributive lattice the Alexander dual is computed.
We provide some new conditions under which the graded Betti numbers of a monomial ideal can be computed in terms of the graded Betti numbers of smaller ideals, thus complementing Eliahou and Kervaire's splitting approach. As applications,…
We use the idea of generic extensions to investigate the correspondence between the isomorphism classes of nilpotent representations of a cyclic quiver and the orbits in the corresponding representation varieties. We endow the set $\cal M$…
We consider an homogeneous ideal $I$ in the polynomial ring $S=K[x_1,\dots,$ $x_m]$ over a finite field $K=\mathbb{F}_q$ and the finite set of projective rational points $\mathbb{X}$ that it defines in the projective space…
Cellular resolutions are a technique for constructing resolutions of monomial ideals by giving a cell complex labeled by monomials, or more generally, by monomial modules. This \verb|Macaulay2| package allows us to work with cellular…
To a natural number $n$, a finite partially ordered set $P$ and a poset ideal ${\mathcal J}$ in the poset $Hom(P,[n])$ of isotonian maps from $P$ to the chain on $n$ elements, we associate two monomial ideals, the letterplace ideal…
We explore connections between the approach of solving the rational interpolation problem via resolutions of ideals and syzygies with the standard method provided by the Extended Euclidean Algorithm. As a consequence, we obtain explicit…
We define the Buchberger resolution, which is a graded free resolution of a monomial ideal in a polynomial ring. Its construction uses a generalization of the Buchberger graph and encodes much of the combinatorics of the Buchberger…
For a partition $\lambda$ of $n \in \mathbb{N}$, let $I^{\rm Sp}_\lambda$ be the ideal of $R=K[x_1,\ldots,x_n]$ generated by all Specht polynomials of shape $\lambda$. We assume that ${\rm char}(K)=0$. Then $R/I^{\rm Sp}_{(n-2,2)}$ is…
When a monomial ideal has linear quotients with respect to an admissible order of increasing support-degree, we provide two proofs of different flavors to show that it is componentwise support-linear. We also introduce the variable…
Let $S$ be a polynomial algebra over a field. We study classes of monomial ideals (as for example lexsegment ideals) of $S$ having minimal depth. In particular, Stanley's conjecture holds for these ideals. Also we show that if Stanley's…
Every quotient R/I of a semigroup ring R by a radical monomial ideal I has a unique minimal injective-like resolution by direct sums of quotients of R modulo prime monomial ideals. The quotient R/I is Cohen-Macaulay if and only if every…
A minor is principal means it is defined by the same row and column indices. Let $X$ be a square generic matrix, $K[X]$ the polynomial ring in entries of $X$, over an algebraically closed field, $K$. For fixed $t\leq n$, let $\mathfrak P_t$…
Let $R:= K[x_1,\ldots,x_{n}]$ be a polynomial ring over an infinite field $K$, and let $I \subset R$ be a homogeneous ideal with respect to a weight vector $\omega = (\omega_1,\ldots,\omega_n) \in (\mathbb{Z}^+)^n$ such that $\dim(R/I) =…
An ideal of polynomials is symmetric if it is closed under permutations of variables. We relate general symmetric ideals to the so called Specht ideals generated by all Specht polynomials of a given shape. We show a connection between the…
With a view to study problems of smoothability, we construct a minimal free resolution for the coordinate ring of an algebroid monomial curve associated to an $AS$ numerical semigroup (i.e. generated by an arithmetic sequence), obtained…
Inspired by a question of Lov\'asz, we introduce a hierarchy of nested semidefinite relaxations of the convex hull of real solutions to an arbitrary polynomial ideal, called theta bodies of the ideal. For the stable set problem in a graph,…
The purpose of this note is to introduce a multiplication on the set of homogeneous polynomials of fixed degree d, in a way to provide a duality theory between monomial ideals of K[x_1,\ldots,x_d] generated in degrees \leq n and block…