Powers of Componentwise Linear Ideals
AbstractWe give criteria for graded ideals to have the property that all their powers are componentwise linear. Typical examples to which our criteria can be applied include the vertex cover ideals of...
View ArticleAlexander duality and resolutions
AbstractChapter 8 begins with Hochster’s formula to compute the graded Betti numbers of Stanley–Reisner ideals and Reisner’s Cohen–Macaulay criterion for simplicial complexes. Then the Eagon–Reiner...
View ArticleHilbert functions and the theorems of Macaulay and Kruskal–Katona
AbstractChapter 6 offers basic material on combinatorics of monomial ideals. First we recall the concepts of Hilbert functions and Hilbert polynomials, and their relationship to the f-vector of a...
View ArticleA short introduction to Gröbner bases
AbstractIn Chapter 2 a short introduction to the main features of Gröbner basis theory is given, including the Buchberger criterion and algorithm. These basic facts are discussed in a comprehensive but...
View ArticleResolutions of monomial ideals and the Eliahou–Kervaire formula
AbstractChapter 7 discusses minimal free resolutions of monomial ideals. We derive formulas for the graded Betti numbers of stable and squarefree stable ideals, and use these formulas to deduce the...
View ArticleThe exterior algebra
AbstractChapter 5 is devoted to establishing the theory of Gröbner bases in the exterior algebra, and uses exterior techniques to give a proof of the Alexander duality theorem which establishes...
View ArticleMonomial orders and weights
AbstractChapter 3 presents one of the most fundamental results on initial ideals, which says that the graded Betti numbers of the initial ideal in<(I) are greater than or equal to the corresponding...
View ArticleMonomial Ideals
AbstractChapter 1 summarizes fundamental material on monomial ideals. In particular, we consider the integral closure of monomial ideals, squarefree normally torsionfree ideals, squarefree monomial...
View ArticleSmooth Fano Polytopes Arising from Finite Partially Ordered Sets
AbstractGorenstein Fano polytopes arising from finite partially ordered sets will be introduced. Then we study the problem of which partially ordered sets yield smooth Fano polytopes.
View ArticleRoots of Ehrhart polynomials arising from graphs
AbstractSeveral polytopes arise from finite graphs. For edge and symmetric edge polytopes, in particular, exhaustive computation of the Ehrhart polynomials not merely supports the conjecture of Beck et...
View ArticleA Quick Introduction to Gröbner Bases
AbstractNeither specialist knowledge nor extensive investment of time is required in order for a nonspecialist to learn fundamentals on Gröbner bases. The purpose of this chapter is to provide the...
View ArticleKoszul Binomial Edge Ideals
AbstractIt is shown that if the binomial edge ideal of a graph G defines a Koszul algebra, then G must be chordal and claw free. A converse of this statement is proved for a class of chordal and...
View ArticleLinearly related polyominoes
AbstractWe classify all convex polyomino ideals which are linearly related or have a linear resolution. Convex stack polyominoes whose ideals are extremal Gorenstein are also classified. In addition,...
View ArticleChain Polytopes and Algebras with Straightening Laws
AbstractIt will be shown that the toric ring of the chain polytope of a finite partially ordered set is an algebra with straightening laws on a finite distributive lattice. Thus, in particular, every...
View ArticleDominating induced matchings of finite graphs and regularity of edge ideals
AbstractThe regularity of the edge ideal of a finite simple graph G is at least the induced matching number of G and is at most the minimum matching number of G. If G possesses a dominating induced...
View ArticleStrongly Koszul Edge Rings
AbstractWe classify the finite connected simple graphs whose edge rings are strongly Koszul. From the classification, it follows that if the edge ring is strongly Koszul, then its toric ideal possesses...
View ArticleInteger Decomposition Property of Free Sums of Convex Polytopes
AbstractLet \({\mathcal{P} \subset \mathbb{R}^{d}}\) and \({\mathcal{Q} \subset \mathbb{R}^{e}}\) be integral convex polytopes of dimension d and e which contain the origin of \({\mathbb{R}^{d}}\) and...
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