Related papers: Vanishing ideals of Lattice Diagram determinants
The aim of this work is to study some lattice diagram determinants $\Delta_L(X,Y)$. We recall that $M_L$ denotes the space of all partial derivatives of $\Delta_L$. In this paper, we want to study the space $M^k_{i,j}(X,Y)$ which is defined…
A lattice diagram is a finite list L=((p_1,q_1),...,(p_n,q_n) of lattice cells. The corresponding lattice diagram determinant is \Delta_L(X;Y)=\det \| x_i^{p_j}y_i^{q_j} \|. These lattice diagram determinants are crucial in the study of the…
The aim of this work is to study some lattice diagram polynomials $\Delta_D(X,Y)$. We recall that $M_D$ denotes the space of all partial derivatives of $\Delta_D$. In this paper, we want to study the space $M^k_{i,j}(X,Y)$ which is the sum…
The space $M_{\mu/i,j}$ spanned by all partial derivatives of the lattice polynomial $\Delta_{\mu/i,j}(X;Y)$ is investigated in math.CO/9809126 and many conjectures are given. Here, we prove all these conjectures for the $Y$-free component…
The lattice cell in the ${i+1}^{st}$ row and ${j+1}^{st}$ column of the positive quadrant of the plane is denoted $(i,j)$. If $\mu$ is a partition of $n+1$, we denote by $\mu/ij$ the diagram obtained by removing the cell $(i,j)$ from the…
A distributive lattice $L$ with minimum element $0$ is called decomposable lattice if $a$ and $b$ are not comparable elements in $L$ there exist $\overline{a},\overline{b}\in L$ such that $a=\overline{a}\vee(a\wedge b),…
Let $\K$ be a finite field and $X$ be a complete simplicial toric variety over $\K$. We give an algorithm relying on elimination theory for finding generators of the vanishing ideal of a subgroup $Y_Q$ parameterized by a matrix $Q$ which…
A distributive lattice structure ${\mathbf M}(G)$ has been established on the set of perfect matchings of a plane bipartite graph $G$. We call a lattice {\em matchable distributive lattice} (simply MDL) if it is isomorphic to such a…
Let K be a finite field with q elements and let X be a subset of a projective space P^{s-1}, over the field K, which is parameterized by Laurent monomials. Let I(X) be the vanishing ideal of X. Some of the main contributions of this paper…
We study the regularity and the algebraic properties of certain lattice ideals. We establish a map I --> I\~ between the family of graded lattice ideals in an N-graded polynomial ring over a field K and the family of graded lattice ideals…
Let $X$ be a complete simplicial toric variety over a finite field $\mathbb{F}_q$ with homogeneous coordinate ring $S=\mathbb{F}_q[x_1,\dots,x_r]$ and split torus $T_X\cong (\mathbb{F}^*_q)^n$. We prove that vanishing ideal of a subset $Y$…
Let K be a field and let M_n(K) denote the space of n x n matrices with entries in K. Let M be a subspace of M_n(K) of dimension d with the property that there are elements in M with non-zero determinant. Given a basis of M, we define the…
A distributive lattice $L$ with minimum element $0$ is called decomposable if $a$ and $b$ are not comparable elements in $L$ then there exist $\overline{a},\overline{b}\in L$ such that $a=\overline{a}\vee(a\wedge b),…
Given a square-free monomial ideal $I$ in a polynomial ring $R$ over a field $\mathbb{K}$, one can associate it with its LCM-lattice and its hypergraph. In this short note, we establish the connection between the LCM-lattice and the…
In this paper we shall consider the Lie algebra of column-finite infinite matrices indexed by positive integers $\mathbb{N}$, describe the lattice of its ideals for arbitrary field $K$ and study its derivations over any commutative, unital…
The main ingredient to construct an O-border basis of an ideal I $\subseteq$ K[x1,. .., xn] is the order ideal O, which is a basis of the K-vector space K[x1,. .., xn]/I. In this paper we give a procedure to find all the possible order…
Let $X$ be an $(m\times n)$-matrix of indeterminates, and let $J$ be the ideal generated by a set $\mathcal{S}$ of maximal minors of $X$. We construct the linear strand of the resolution of $J$. This linear strand is determined by the…
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.
In this paper we primarily study monomial ideals and their minimal free resolutions by studying their associated LCM lattices. In particular, we formally define the notion of coordinatizing a finite atomic lattice P to produce a monomial…
Infinite hyperplane arrangements whose vertices form a lattice are studied from the point of view of commutative algebra. The quotient of such an arrangement modulo the lattice action represents the minimal free resolution of the associated…