Related papers: Markov bases and generalized Lawrence liftings
Any configuration of lattice vectors gives rise to a hierarchy of higher-dimensional configurations which generalize the Lawrence construction in geometric combinatorics. We prove finiteness results for the Markov bases, Graver bases and…
We discuss connecting tables with zero-one entries by a subset of a Markov basis. In this paper, as a Markov basis we consider the Graver basis, which corresponds to the unique minimal Markov basis for the Lawrence lifting of the original…
In this paper we study the computation of Markov bases for contingency tables whose cell entries have an upper bound. In general a Markov basis for unbounded contingency table under a certain model differs from a Markov basis for bounded…
We present a new algorithm for computing a truncated Markov basis of a lattice. In general, this new algorithm is faster than existing methods. We then extend this new algorithm so that it solves the linear integer feasibility problem with…
We study how to lift Markov bases and Gr\"obner bases along linear maps of lattices. We give a lifting algorithm that allows to compute such bases iteratively provided a certain associated semigroup is normal. Our main application is the…
To any integer matrix $A$ one can associate a matroid structure consisting of a graph and another integer matrix $A_B$. The connected components of this graph are called bouquets. We prove that bouquets behave well with respect to the…
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…
Motivated by the behavior of the trace pairing over tame cyclic number fields, we introduce the notion of tame lattices. Given an arbitrary non-trivial lattice $\mathcal{L}$ we construct a parametric family of full-rank sub-lattices…
We show that the complexity of the Markov bases of multidimensional tables stabilizes eventually if a single table dimension is allowed to vary. In particular, if this table dimension is beyond a computable bound, the Markov bases consist…
In this paper, we draw a connection between ideal lattices and Gr\"{o}bner bases in the multivariate polynomial rings over integers. We study extension of ideal lattices in $\mathbb{Z}[x]/\langle f \rangle$ (Lyubashevsky \& Micciancio,…
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…
Blockwise determinantal ideals are those generated by the union of all the minors of specified sizes in certain blocks of a generic matrix, and they are the natural generalization of many existing determinantal ideals like the Schubert and…
In this paper, we draw connections between ideal lattices and multivariate polynomial rings over integers using Gr\"obner bases. Ideal lattices are ideals in the residue class ring, $\mathbb{Z}[x]/\langle f \rangle$ (here $f$ is a monic…
Link et al. (2010) define a general framework for analyzing capture-recapture data with potential misidentifications. In this framework, the observed vector of counts, $y$, is considered as a linear function of a vector of latent counts,…
We determine an explicit Gr\"obner basis, consisting of linear forms and determinantal quadrics, for the prime ideal of Raftery's mixture transition distribution model for Markov chains. When the states are binary, the corresponding…
In this paper, we give a sufficient condition for a set $\mathal G$ of polynomials to be a Gr\"obner basis with respect to a given term-order for the ideal $I$ that it generates. Our criterion depends on the linkage pattern of the ideal $I$…
In a probabilistic graphical model on a set of variables $V$, the Markov blanket of a random vector $B$ is the minimal set of variables conditioned to which $B$ is independent from the remaining of the variables $V \backslash B$. We…
In this paper, we show that for each lattice basis, there exists an equivalent basis which we describe as ``strongly reduced''. We show that bases reduced in this manner exhibit rather ``short'' basis vectors, that is, the length of the…
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…
Gr\"obner bases of binomial ideals arising from finite lattices will be studied. In terms of Gr\"obner bases and initial ideals, a characterization of finite distributive lattices as well as planar distributive lattices will be given.