Empty Simplices of Polytopes and Graded Betti Numbers

2008 ◽  
pp. 1-22
Author(s):  
Uwe Nagel
Keyword(s):  
Betti Numbers ◽  
Graded Betti Numbers

Weakly Polymatroidal Ideals

2006 ◽  
Vol 13 (04) ◽  
pp. 711-720 ◽  
Author(s):  
Masako Kokubo ◽  
Takayuki Hibi
Keyword(s):  
Betti Numbers ◽  
Partially Ordered Sets ◽  
Finite Sequence ◽  
Ordered Sets ◽  
Partially Ordered ◽  
Graded Betti Numbers ◽  
Linear Resolution ◽  
Fundamental Fact ◽  
Polymatroidal Ideals ◽  
Polymatroidal Ideal

The concept of the weakly polymatroidal ideal, which generalizes both the polymatroidal ideal and the prestable ideal, is introduced. A fundamental fact is that every weakly polymatroidal ideal has a linear resolution. One of the typical examples of weakly polymatroidal ideals arises from finite partially ordered sets. We associate each weakly polymatroidal ideal with a finite sequence, alled the polymatroidal sequence, which will be useful for the computation of graded Betti numbers of weakly polymatroidal ideals as well as for the construction of weakly polymatroidal ideals.

scholarly journals The regularity of Tor and graded Betti Numbers

2006 ◽  
Vol 128 (3) ◽  
pp. 573-605 ◽  
Author(s):  
David Eisenbud ◽  
C. (Craig) Huneke ◽  
Bernd Ulrich
Keyword(s):  
Betti Numbers ◽  
Graded Betti Numbers

On Graded Betti Numbers of a Class of Graphs

2018 ◽  
Vol 25 (02) ◽  
pp. 335-348 ◽  
Author(s):  
Saba Yasmeen ◽  
Tongsuo Wu
Keyword(s):  
Betti Numbers ◽  
Bipartite Graphs ◽  
Complete Graphs ◽  
Graded Betti Numbers

In this paper, Betti numbers are evaluated for several classes of graphs whose complements are bipartite graphs. Relations are established for the general case, and counting formulae are given in several particular cases, including the union of several mutually disjoint complete graphs.

scholarly journals Path ideals of rooted trees and their graded Betti numbers

2011 ◽  
Vol 118 (8) ◽  
pp. 2411-2425 ◽  
Author(s):  
Rachelle R. Bouchat ◽  
Huy Tài Hà ◽  
Augustine OʼKeefe
Keyword(s):  
Betti Numbers ◽  
Rooted Trees ◽  
Graded Betti Numbers

Projective curves of degree=codimension+2 II

2016 ◽  
Vol 26 (01) ◽  
pp. 95-104 ◽  
Author(s):  
Wanseok Lee ◽  
Euisung Park
Keyword(s):  
Integral Curve ◽  
Betti Numbers ◽  
Free Resolution ◽  
Relative Location ◽  
Normal Curve ◽  
Minimal Free Resolution ◽  
Graded Betti Numbers ◽  
Projective Curves ◽  
Rational Normal Curve

Let [Formula: see text] be a nondegenerate projective integral curve of degree [Formula: see text] which is not linearly normal. In this paper, we continues the study begun in [E. Park, Projective curves of degree=codimension+2, Math. Z. 256 (2007) 685–697] for the minimal free resolution of [Formula: see text]. It is well-known that [Formula: see text] is an isomorphic projection of a rational normal curve [Formula: see text] from a point [Formula: see text]. Our main result is about how the graded Betti numbers of [Formula: see text] are determined by the rank of [Formula: see text] with respect to [Formula: see text], which is a measure of the relative location of [Formula: see text] from [Formula: see text].

scholarly journals Smallest Graded Betti Numbers

2001 ◽  
Vol 244 (1) ◽  
pp. 236-259 ◽  
Author(s):  
Benjamin P Richert
Keyword(s):  
Betti Numbers ◽  
Graded Betti Numbers

scholarly journals Algebraic shifting and graded Betti numbers

2008 ◽  
Vol 361 (04) ◽  
pp. 1853-1865 ◽  
Author(s):  
Satoshi Murai ◽  
Takayuki Hibi
Keyword(s):  
Betti Numbers ◽  
Graded Betti Numbers ◽  
Algebraic Shifting

scholarly journals Componentwise linear ideals

1999 ◽  
Vol 153 ◽  
pp. 141-153 ◽  
Author(s):  
Jürgen Herzog ◽  
Takayuki Hibi
Keyword(s):  
Simplicial Complex ◽  
Polynomial Ring ◽  
Betti Numbers ◽  
Monomial Ideals ◽  
Homogeneous Polynomials ◽  
Graded Betti Numbers ◽  
Linear Resolution ◽  
Alexander Dual ◽  
Componentwise Linear Ideals ◽  
The Ideal

AbstractA componentwise linear ideal is a graded ideal I of a polynomial ring such that, for each degree q, the ideal generated by all homogeneous polynomials of degree q belonging to I has a linear resolution. Examples of componentwise linear ideals include stable monomial ideals and Gotzmann ideals. The graded Betti numbers of a componentwise linear ideal can be determined by the graded Betti numbers of its components. Combinatorics on squarefree componentwise linear ideals will be especially studied. It turns out that the Stanley-Reisner ideal IΔ arising from a simplicial complex Δ is componentwise linear if and only if the Alexander dual of Δ is sequentially Cohen-Macaulay. This result generalizes the theorem by Eagon and Reiner which says that the Stanley-Reisner ideal of a simplicial complex has a linear resolution if and only if its Alexander dual is Cohen-Macaulay.

scholarly journals Betti numbers of toric ideals of graphs: A case study

2019 ◽  
Vol 18 (12) ◽  
pp. 1950226
Author(s):  
Federico Galetto ◽  
Johannes Hofscheier ◽  
Graham Keiper ◽  
Craig Kohne ◽  
Adam Van Tuyl ◽  
...  
Keyword(s):  
Bipartite Graph ◽  
Hilbert Series ◽  
Complete Bipartite Graph ◽  
Betti Numbers ◽  
Toric Ideals ◽  
Toric Ideal ◽  
Initial Ideal ◽  
Graded Betti Numbers ◽  
Linear Quotients

We compute the graded Betti numbers for the toric ideal of a family of graphs constructed by adjoining a cycle to a complete bipartite graph. The key observation is that this family admits an initial ideal which has linear quotients. As a corollary, we compute the Hilbert series and [Formula: see text]-vector for all the toric ideals of graphs in this family.

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