Related papers: On the Gr\"obner complexity of matrices

Let $\mathcal{A}=\{{\bf a}_1,\ldots,{\bf a}_n\}\subset\Bbb{N}^m$. We give an algebraic characterization of the universal Markov basis of the toric ideal $I_{\mathcal{A}}$. We show that the Markov complexity of $\mathcal{A}=\{n_1,n_2,n_3\}$…

We investigate the computational complexity of problems on toric ideals such as normal forms, Gr\"obner bases, and Graver bases. We show that all these problems are strongly NP-hard in the general case. Nonetheless, we can derive efficient…

Associated to any vector configuration A is a toric ideal encoded by vectors in the kernel of A. Each toric ideal has two special generating sets: the universal Gr\"obner basis and the Graver basis. While the former is generally a proper…

In this paper, we present a construction that turns certain relations on Graver basis elements of an $M$-fold matrix $A^{(M)}$ into relations on Graver basis elements of an $(M+1)$-fold matrix $A^{(M+1)}$. In doing so, we strengthen the…

The universal Gr\"obner basis of an ideal is a Gr\"obner basis with respect to all term orders simultaneously. The aim of this paper is to present an algorithmic approach to compute the universal Gr\"obner basis for the toric ideal…

The universal Gr\"{o}bner basis of $I$, is a Gr\"{o}bner basis for $I$ with respect to all term orders simultaneously. Let $I_G$ be the toric ideal of a graph $G$. We characterize in graph theoretical terms the elements of the universal…

The toric ideals of $3\times 3$ transportation polytopes $T$ are quadratically generated. The only exception is the Birkhoff polytope $B_3$. If $T$ is not a multiple of $B_3$, these ideals even have squarefree quadratic initial ideals. This…

Two correspondences have been provided that associate any linear code over a finite field with a binomial ideal. In this paper, algorithms for computing their Graver bases and universal Gr\"obner bases are given. To this end, a connection…

In this article we establish an exponential lower bound on the Graver complexity of integer programs. This provides new type of evidence supporting the presumable intractability of integer programming. Specifically, we show that the Graver…

Let $I_1\subset I_2\subset\dots$ be an increasing sequence of ideals of the ring $\Bbb Z[X]$, $X=(x_1,\dots,x_n)$ and let $I$ be their union. We propose an algorithm to compute the Gr\"obner base of $I$ under the assumption that the…

Each linear code can be described by a code ideal given as the sum of a toric ideal and a non-prime ideal. In this way, several concepts from the theory of toric ideals can be translated into the setting of code ideals. It will be shown…

We apply the theory of Groebner bases to the study of signed, symmetric polyomino tilings of planar domains. Complementing the results of Conway and Lagarias we show that the triangular regions T_N=T_{3k-1} and T_N=T_{3k} in a hexagonal…

Let $A(n,r;3)$ be the total weight of the alternating sign matrices of order $n$ whose sole `1' of the first row is at the $r^{th}$ column and the weight of an individual matrix is $3^k$ if it has $k$ entries equal to -1. Define the…

Solving a polynomial system, or computing an associated Gr\"obner basis, has been a fundamental task in computational algebra. However, it is also known for its notorious doubly exponential time complexity in the number of variables in the…

Let C be a uniform clutter, i.e., all the edges of C have the same size, and let A be the incidence matrix of C. We denote the column vectors of A by v1,...,vq. The vertex covering number of C, denoted by g, is the smallest number of…

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)…

We prove the so-called inverse conjecture for the Gowers U^{s+1}-norm in the case s = 3 (the cases s < 3 being established in previous literature). That is, we establish that if f : [N] -> C is a function with |f(n)| <= 1 for all n and || f…

The toric ring together with the toric ideal arising from a nested configuration is studied, with particular attention given to the algebraic study of normality of the toric ring as well as the Gr\"obner bases of the toric ideal. One of the…

To integer programming problems, computational algebraic approaches using Grobner bases or standard pairs via the discreteness of toric ideals have been studied in recent years. Although these approaches have not given improved time…

In this paper we completely describe the unital compressed commuting graph of the ring $\mathcal{M}_3(\mathrm{GF}(p))$ of $3 \times 3$ matrices over the finite prime field $\mathrm{GF}(p)$. To achieve this we combine methods from linear…