Toric Ideals Generated by Quadratic Binomials
Hidefumi Ohsugi,Takayuki Hibi +1 more
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In this article, a combinatorial criterion for the toric ideal arising from a finite graph to be generated by quadratic binomials is studied, and it is shown that every normal non-Koszul semigroup ring generated by square-free quadratically monomials has a 2-linear resolution.About:
This article is published in Journal of Algebra.The article was published on 1999-08-15 and is currently open access. It has received 217 citations till now. The article focuses on the topics: Ideal (ring theory) & Koszul algebra.read more
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Monomial ideals whose powers have a linear resolution
TL;DR: In this paper, the authors consider graded ideals in a polynomial ring over a field and ask when such an ideal has the property that all of its powers have a linear resolution.
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Grobner Bases in Commutative Algebra
Viviana Ene,Jürgen Herzog +1 more
TL;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.
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Monomial ideals whose powers have a linear resolution
TL;DR: In this paper, it was shown that all powers of a monomial ideal with 2-linear resolution have a linear resolution, and that all the powers of monomial ideals with 3-linear resolutions have linear resolutions.
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Four Counterexamples in Combinatorial Algebraic Geometry
TL;DR: In this paper, the authors present counterexamples to four conjectures which appeared in the literature in com-mutative algebra and algebraic geometry. The four questions are largely unrelated, and yet their answers are connected by a common thread: they are combina-torial in nature, involving monomial ideals and binomial ideals, and they were found by exhaustive computer search using the symbolic algebra systems Maple and Macaulay 2.
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Ideals, Varieties, and Algorithms
TL;DR: In the Groebner package, the most commonly used commands are NormalForm, for doing the division algorithm, and Basis, for computing a Groebners basis as mentioned in this paper. But these commands require a large number of variables.
Book
Cohen-Macaulay rings
Winfried Bruns,H. Jürgen Herzog +1 more
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.
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Gröbner bases and convex polytopes
TL;DR: Grobner basics The state polytope Variation of term orders Toric ideals Enumeration, sampling and integer programming Primitive partition identities Universal Grobner bases Regular triangulations The second hypersimplex $\mathcal A$-graded algebras Canonical subalgebra bases Generators, Betti numbers and localizations Toric varieties in algebraic geometry as mentioned in this paper.
Book
Gröbner Bases: A Computational Approach to Commutative Algebra
TL;DR: This chapter discusses linear algebra in Residue Class Rings in Vector Spaces and Modules, and first applications of Gr bner Bases.