Ideals of adjacent minors
TL;DR: In this article, the authors give a description of the minimal primes of the ideal generated by the 2×2 adjacent minors of a generic matrix and show that the ideals which appear as minimal prime ideals are, in fact, prime ideals.
About: This article is published in Journal of Algebra.The article was published on 2004-07-15 and is currently open access. It has received 53 citations till now. The article focuses on the topics: Boolean prime ideal theorem & Minimal ideal.
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TL;DR: It follows that all binomial edge ideals are radical ideals, and the results apply for the class of conditional independence ideals where a fixed binary variable is independent of a collection of other variables, given the remaining ones.
244 citations
TL;DR: In this article, the Grobner basis of I G of S generated by 2-minors [i, j] of X which correspond to edges of G was constructed.
Abstract: Let G be a finite graph on [n] = {1, 2,…, n}, X a 2 × n matrix of indeterminates over a field K, and S = K[X] a polynomial ring over K. In this article, we study about ideals I G of S generated by 2-minors [i, j] of X which correspond to edges {i, j} of G. In particular, we construct a Grobner basis of I G as a set of paths of G and compute a primary decomposition.
156 citations
Cites background from "Ideals of adjacent minors"
...Hoşten and Sullivant [7] studied ideals of adjacent minors as a generalization of ideals of Diaconis, Eisenbud, and Sturmfels....
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...Hoşten and Sullivant [7] studied ideals of adjacent minors as a generalization of ideals of Diaconis, Eisenbud, and Sturmfels....
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TL;DR: This tutorial will consist of a detailed study of two examples where the algebra/statistics connection has proven especially useful: in the study of phylogenetic models and in the analysis of contingency tables.
Abstract: Algebraic statistics advocates polynomial algebra as a tool for addressing problems in statistics and its applications. This connection is based on the fact that most statistical models are defined either parametrically or implicitly via polynomial equations. The idea is summarized by the phrase "Statistical models are semialgebraic sets". My tutorial will consist of a detailed study of two examples where the algebra/statistics connection has proven especially useful: in the study of phylogenetic models and in the analysis of contingency tables.
155 citations
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01 Dec 2011TL;DR: Polynomial rings and ideals Grobner bases as mentioned in this paper have been applied in commutative algebra and combinatorics, and they have been used for modules and toric ideals.
Abstract: Polynomial rings and ideals Grobner bases First applications Grobner bases for modules Grobner bases of toric ideals Selected applications in commutative algebra and combinatorics Bibliography Index
133 citations
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TL;DR: In this article, the authors introduced binomial edge ideals attached to a simple graph and studied their algebraic properties, and provided sufficient conditions for Cohen-Macaulayness for closed and non-closed graphs.
Abstract: We introduce binomial edge ideals attached to a simple graph $G$ and study their algebraic properties. We characterize those graphs for which the quadratic generators form a Gr\"obner basis in a lexicographic order induced by a vertex labeling. Such graphs are chordal and claw-free. We give a reduced squarefree Gr\"obner basis for general $G$. It follows that all binomial edge ideals are radical ideals. Their minimal primes can be characterized by particular subsets of the vertices of $G$. We provide sufficient conditions for Cohen--Macaulayness for closed and nonclosed graphs.
Binomial edge ideals arise naturally in the study of conditional independence ideals. Our results apply for the class of conditional independence ideals where a fixed binary variable is independent of a collection of other variables, given the remaining ones. In this case the primary decomposition has a natural statistical interpretation
132 citations
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01 Jan 1993
TL;DR: In this article, the authors present a self-contained introduction to the homological and combinatorial aspects of the theory of Cohen-Macaulay rings, Gorenstein rings, local cohomology, and canonical modules.
Abstract: In the last two decades Cohen-Macaulay rings and modules have been central topics in commutative algebra. This book meets the need for a thorough, self-contained introduction to the homological and combinatorial aspects of the theory of Cohen-Macaulay rings, Gorenstein rings, local cohomology, and canonical modules. A separate chapter is devoted to Hilbert functions (including Macaulay's theorem) and numerical invariants derived from them. The authors emphasize the study of explicit, specific rings, making the presentation as concrete as possible. So the general theory is applied to Stanley-Reisner rings, semigroup rings, determinantal rings, and rings of invariants. Their connections with combinatorics are highlighted, e.g. Stanley's upper bound theorem or Ehrhart's reciprocity law for rational polytopes. The final chapters are devoted to Hochster's theorem on big Cohen-Macaulay modules and its applications, including Peskine-Szpiro's intersection theorem, the Evans-Griffith syzygy theorem, bounds for Bass numbers, and tight closure. Throughout each chapter the authors have supplied many examples and exercises which, combined with the expository style, will make the book very useful for graduate courses in algebra. As the only modern, broad account of the subject it will be essential reading for researchers in commutative algebra.
2,760 citations
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01 Jan 1978
TL;DR: The second edition has been updated and revised, with more emphasis on logic and logistic response properties and on the small-sample behavior of chi-square statistics as mentioned in this paper, and includes 40 to 50 new problems with most having separate data sheets.
Abstract: The second edition has been updated and revised, with more emphasis on logic and logistic response properties and on the small-sample behavior of chi-square statistics. It includes 40 to 50 new problems with most having separate data sheets. A solutions manual is available to qualified instructors. In addition, some minor revisions in early chapters have been added to clarify such terms as order categories and collapsing.
2,379 citations
12 Sep 2002
TL;DR: Polynomials in one variable Grobner bases of zero-dimensional ideals Bernstein's theorem and fewnomials as mentioned in this paper are the primary decomposition of polynomial systems in economics and statistics.
Abstract: Polynomials in one variable Grobner bases of zero-dimensional ideals Bernstein's theorem and fewnomials Resultants Primary decomposition Polynomial systems in economics Sums of squares Polynomial systems in statistics Tropical algebraic geometry Linear partial differential equations with constant coefficients Bibliography Index.
860 citations
"Ideals of adjacent minors" refers background in this paper
... probability distributions of these random variables (the so-called no dway interaction models [8]) gives rise to a toric variety whose set of defining equations may be extremely large and complicated [1, 15]. However, the positive probability distributions are described precisely by the simple multidimensional adjacent minors we will introduce. The story of the minimal primes of these ideals is far from ...
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TL;DR: The second edition has been updated and revised, with more emphasis on logic and logistic response properties and on the small-sample behavior of chi-square statistics.
573 citations