Related papers: Finite lattices and Gr\"obner bases
Minimal Markov bases of configurations of integer vectors correspond to minimal binomial generating sets of the assocciated lattice ideal. We give necessary and sufficient conditions for the elements of a minimal Markov basis to be (a)…
In this paper, we study ideals spanned by polynomials or overconvergent series in a Tate algebra. With state-of-the-art algorithms for computing Tate Gr{\"o}bner bases, even if the input is polynomials, the size of the output grows with the…
In this paper, a method for constructing a near optimal normal basis for algebraic extensions of a finite field is described. In each extension, except for the squares of basis elements, the product of two distinct normal basis elements can…
Using recent work by Erman-Sam-Snowden, we show that finitely generated ideals in the ring of bounded-degree formal power series in infinitely many variables have finitely generated Gr\"obner bases relative to the graded reverse…
From algebraic geometry perspective database relations are succinctly defined as Finite Varieties. After establishing basic framework, we give analytic proof of Heath theorem from Database Dependency theory. Next, we leverage…
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…
We give an explicit upper bound for the degree of a tropical basis of a homogeneous polynomial ideal. As an application f-vectors of tropical varieties are discussed. Various examples illustrate differences between Gr\"obner and tropical…
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 introduce and study the concept of positive polynomial ideals between Banach lattices. The paper develops the basic principles of these classes and presents methods for constructing positive polynomial ideals from given positive operator…
For a partition $\lambda$ of $n$, the _Specht ideal_ $I_\lambda \subset K[x_1, \ldots, x_n]$ is the ideal generated by all Specht polynomials of shape $\lambda$. In their unpublished manuscript, Haiman and Woo showed that $I_\lambda$ is a…
We give the generating function for the index of integer lattice points, relative to a finite order ideal. The index is an important concept in the theory of border bases, an alternative to Gr\"obner bases. Equivalently, we explicitly solve…
Dilworth's theorem. Every finite distributive lattice $D$ can be represented as the congruence lattice of a finite lattice $L$. We want: Every finite distributive lattice $D$ can be represented as the congruence lattice of a nice finite…
Our main goal is to develop a representation for finite distributive nearlattices through certain ordered structures. This representation generalizes the well-known representation given by Birkhoff for finite distributive lattices through…
There is an extensive recent literature on the graded, non-graded, prime, primitive, maximal ideals of Leavitt path algebras. In this introductory level survey, we will be giving an overview of different types of ideals and the…
Parametric Gr\"obner bases have been studied for more than 15 years and are now a further developed subject. Here we propose a general study of parametric standard bases, that is with local orders. We mainly focus on the commutative case…
Let $L\subset \mathbb{Z}^n$ be a lattice and $I_L=\langle x^{\bf u}-x^{\bf v}:\ {\bf u}-{\bf v}\in L\rangle$ be the corresponding lattice ideal in $\Bbbk[x_1,\ldots, x_n]$, where $\Bbbk$ is a field. In this paper we describe minimal…
This note computes a Gr\"obner basis for the ideal defining a union of Schubert varieties. More precisely, it computes a Gr\"obner basis for unions of schemes given by northwest rank conditions on the space of all matrices of a fixed size.…
We consider the problem of determining Gr\"obner bases of binomial ideals associated with linear error correcting codes. Computation of Gr\"obner bases of linear codes have become a topic of interest to many researchers in coding theory…
There is a natural infinite graph whose vertices are the monomial ideals in a polynomial ring. The definition involves Gr\"obner bases or the action of an algebraic torus. We present algorithms for computing the (affine schemes…
This paper presents a conception for computing gr\"{o}bner basis. We convert some of gr\"{o}bner-computing algorithms, e.g., F5, extended F5 and GWV algorithms into a special type of algorithm. The new algorithm's finite termination problem…