Related papers: Groebner bases of ideals invariant under endomorph…
In this expositional paper, we discuss commutative algebra -- a study inspired by the properties of integers, rational numbers, and real numbers. In particular, we investigate rings and ideals, and their various properties. After, we…
In this article we produce Groebner bases for the defining ideal of a monomial curve that corresponds to an almost arithmetic sequence of positive integers, correcting previous work of Sengupta,(2003).
In this paper, we define an invariant, which we believe should be the substitute for total K-theory in the case when there is one distinguished ideal. Moreover, some diagrams relating the new groups to the ordinary K-groups with…
Given a finite, simple, vertex-weighted graph, we construct a graded associative (non-commutative) algebra, whose generators correspond to vertices and whose ideal of relations has generators that are graded commutators corresponding to…
In this paper we introduce a new and large family of configurations whose toric ideals possess quadratic Groebner bases. As an application, a generalization of algebras of Segre-Veronese type will be studied.
It has been shown previously that a large class of monomial maps equivariant under the action of an infinite symmetric group have finitely generated kernels up to the symmetric action. We prove that these symmetric toric ideals also have…
The goal of this work is to study the ideals of the Goldman Lie algebra $S$. To do so, we construct an algebra homomorphism from $S$ to a simpler algebraic structure, and focus on finding ideals of this new structure instead. The structure…
We establish Gr\"obner--Shirshov bases theory for commutative dialgebras. We show that for any ideal $I$ of $Di[X]$, $I$ has a unique reduced Gr\"obner--Shirshov basis, where $Di[X]$ is the free commutative dialgebra generated by a set $X$,…
In this paper, we elaborate Gr\"obner-Shirshov bases method for Leibniz (super)algebras. We show that there is a unique reduced Gr\"obner-Shirshov basis for every (graded) ideal of a free Leibniz (super)algebra. As applications, we…
We give a criterion for a collection of polynomials to be a universal Gr\"{o}bner basis for an ideal in terms of the multidegree of the closure of the corresponding affine variety in $(\mathbb{P}^1)^N$. This criterion can be used to give…
In this paper, we consider parametric ideals and introduce a notion of comprehensive involutive system. This notion plays the same role in theory of involutive bases as the notion of comprehensive Groebner system in theory of Groebner…
We give an introduction to the theory of initial ideals and initial algebras with emphasis on the transfer of structural properties.
A universal Gr\"obner basis of an ideal is the union of all its reduced Gr\"obner bases. It is contained in the Graver basis, the set of all primitive elements. Obtaining an explicit description of either of these sets, or even a sharp…
In combinatorial commutative algebra and algebraic statistics many toric ideals are constructed from graphs. Keeping the categorical structure of graphs in mind we give previous results a more functorial context and generalize them by…
In classical invariant theory, the Gr\"obner base of the ideal of syzygies and the normal forms of polynomials of invariants are two core contents. To improve the performance of invariant theory in symbolic computing of classical geometry,…
Let $G$ be a split semisimple linear algebraic group over a field and let $X$ be a generic twisted flag variety of $G$. Extending the Hilbert basis techniques to Laurent polynomials over integers we give an explicit presentation of the…
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…
Given a finite-dimensional complex Lie algebra g equipped with a nondegenerate, symmetric, invariant bilinear form B, let V_k(g,B) denote the universal affine vertex algebra associated to g and B at level k. For any reductive group G of…
Given an ample action of an inverse semigroup on a locally compact and Hausdorff topological space, we study the ideal structure of the crossed product algebra associated with it. By developing a theory of induced ideals, we manage to prove…
An algebra $\cal{R}$ is called an extension of the algebra $M$ by $B$ if $M^2=0$, $M$ is an ideal of $\cal{R}$ and $\cal{R}$$/M\cong B$ as algebras. In this paper, by using the Gr\"{o}bner-Shirshov bases, we characterize completely the…