Related papers: Janet Bases of Toric Ideals
In this paper, in addition to the earlier introduced involutive divisions, we consider a new class of divisions induced by admissible monomial orderings. We prove that these divisions are noetherian and constructive. Thereby each of them…
A graph-theoretic method, simpler than existing ones, is used to characterize the minimal set of monomial generators for the integral closure of any algebra of polynomials generated by quadratic monomials. The toric ideal of relations…
Motivated by path-integral numerical solutions of diffusion processes, PATHINT, we present a new tree algorithm, PATHTREE, which permits extremely fast accurate computation of probability distributions of a large class of general nonlinear…
In this article we consider the problem of finding generators of the toric ideal associated to a combinatorial object called a staged tree. Our main theorem states that toric ideals of staged trees that are balanced and stratified are…
We describe an algorithm for the sequential sampling of entries in multiway contingency tables with given constraints. The algorithm can be used for computations in exact conditional inference. To justify the algorithm, a theory relates…
An ideal in a polynomial ring encodes a system of linear partial differential equations with constant coefficients. Primary decomposition organizes the solutions to the PDE. This paper develops a novel structure theory for primary ideals in…
We develop the theory of Gr\"obner bases for ideals in a polynomial ring with countably infinite variables over a field. As an application we reconstruct some of the one-one correspondences among various sets of partitions by using division…
We encode the binomials belonging to the toric ideal $I_A$ associated with an integral $d \times n$ matrix $A$ using a short sum of rational functions as introduced by Barvinok \cite{bar,newbar}. Under the assumption that $d,n$ are fixed,…
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…
Toric codes are obtained by evaluating rational functions of a nonsingular toric variety at the algebraic torus. One can extend toric codes to the so called generalized toric codes. This extension consists on evaluating elements of an…
These notes are based on a series of lectures given by the author at the Max Planck Institute for Mathematics in the Sciences in Leipzig. Addressed topics include affine and projective toric varieties, abstract normal toric varieties from…
In this paper we show that the agglomeration of rows or columns of a contingency table with a hierarchical clustering algorithm yields statistical models defined through toric ideals. In particular, starting from the classical independence…
Border bases can be considered to be the natural extension of Gr\"obner bases that have several advantages. Unfortunately, to date the classical border basis algorithm relies on (degree-compatible) term orderings and implicitly on reduced…
Graph polynomials encode fundamental combinatorial invariants of graphs. Their computation is investigated using tree and path decomposition frameworks, with formal definitions of treewidth, k-trees, and pathwidth establishing the…
This paper presents a novel approach to automatically solving arithmetic word problems. This is the first algorithmic approach that can handle arithmetic problems with multiple steps and operations, without depending on additional…
Given a square matrix $B$ over a principal ideal domain $D$ and an ideal $J$ of $D$, the $J$-ideal of $B$ consists of the polynomials $f\in D[X]$ such that all entries of $f(B)$ are in $J$. It has been shown that in order to determine all…
We illustrate an efficient new method for handling polynomial systems with degenerate solution sets. In particular, a corollary of our techniques is a new algorithm to find an isolated point in every excess component of the zero set (over…
Let I be an arbitrary ideal in C[[x,y]]. We use the Newton algorithm to compute by induction the motivic zeta function of the ideal, yielding only few poles, associated to the faces of the successive Newton polygons. We associate a minimal…
The purpose of this paper is to give an explicit formula which allows one to compute the dimension of the cohomology groups of the sheaf $\Omega_{\P}^p(D)$ of p-th differential forms of Zariski twisted by an ample invertible sheaf on a…
The main theme of this dissertation is the study of the lattice points in a rational convex polyhedron and their encoding in terms of Barvinok's short rational functions. The first part of this thesis looks into theoretical applications of…