相关论文: Vanishing Ideals of Parameterized Subgroups in a t…

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 $X$ be a complete simplicial toric variety over a finite field with a split torus $T_X$. For any matrix $Q$, we are interested in the subgroup $Y_Q$ of $T_X$ parameterized by the columns of $Q$. We give an algorithm for obtaining 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…

Let X be an algebraic toric set in a projective space over a finite field. We study the vanishing ideal, I(X), of X and show some useful degree bounds for a minimal set of generators of I(X). We give an explicit description of a set of…

Motivated by applications to the theory of error-correcting codes, we give methods for computing a generating set for the ideal generated by $\beta$-graded polynomials vanishing on certain subsets of a simplicial complete toric variety $X$…

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 K be a finite field and let X be a subset of a projective space, over the field K, which is parameterized by monomials arising from the edges of a clutter. We show some estimates for the degree-complexity, with respect to the revlex…

We study the vanishing ideal of the parametrized algebraic toric associated to the complete multipartite graph $\G=\mathcal{K}_{\alpha_1,...,\alpha_r}$ over a finite field of order $q$. We give an explicit family of binomial generators for…

We show that the number of rational points of a subgroup inside a toric variety over a finite field defined by a homogeneous lattice ideal can be computed via Smith normal form of the matrix whose columns constitute a basis of the lattice.…

Let $\mathbb{F}_q$ be a finite field, let $\mathbb{X}$ be a subset of a projective space ${\mathbb P}^{s-1}$, over the field $\mathbb{F}_q$, parameterized by rational functions, and let $I(\mathbb{X})$ be the vanishing ideal of…

Let X* be a subset of an affine space A^s, over a finite field K, which is parameterized by the edges of a clutter. Let X and Y be the images of X* under the maps x --> [x] and x --> [(x,1)] respectively, where [x] and [(x,1)] are points in…

A lattice diagram is a finite set $L=\{(p_1,q_1),... ,(p_n,q_n)\}$ of lattice cells in the positive quadrant. The corresponding lattice diagram determinant is $\Delta_L(\X;\Y)=\det \| x_i^{p_j}y_i^{q_j} \|$. The space $M_L$ is the space…

Given a finite set of arbitrarily distributed points in affine space with arbitrary multiplicity structures, we present an algorithm to compute the reduced Groebner basis of the vanishing ideal under the lexicographic ordering. Our method…

We use multiplication maps to give a characteristic-free approach to vanishing theorems on toric varieties. Our approach is very elementary but is enough powerful to prove vanishing theorems.

There are well known algorithms to compute the class group of the maximal order $\mathcal{O}_K$ of a number field $K$ and the group of invertible ideal classes of a non-maximal order $R$. In this paper we explain how to compute also the…

This is an expository paper on Garland's vanishing theorem specialized to the case when the linear algebraic group is $\mathrm{SL}_n$. Garland's theorem can be stated as a vanishing of the cohomology groups of certain finite simplicial…

We study the ideal generated by polynomials vanishing on a semialgebraic set and propose an algorithm to calculate the generators, which is based on some techniques of the cylindrical algebraic decomposition. By applying these, polynomial…

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…

In this paper we characterize, in algebraic and geometric terms, when a graded vanishing ideal is generated by binomials over any field K.

The vanishing ideal of a set of points $X\subseteq \mathbb{R}^n$ is the set of polynomials that evaluate to $0$ over all points $\mathbf{x} \in X$ and admits an efficient representation by a finite set of polynomials called generators. To…