scholarly journals Sharp upper bounds for the Betti numbers of a given Hilbert polynomial

2013 ◽  
Vol 7 (5) ◽  
pp. 1019-1064 ◽  
Author(s):  
Giulio Caviglia ◽  
Satoshi Murai
Keyword(s):  
Betti Numbers ◽  
Upper Bounds ◽  
Hilbert Polynomial

Upper bounds for the betti numbers of graded ideals of a given length in the exterior algebra

1999 ◽  
Vol 27 (9) ◽  
pp. 4607-4631 ◽  
Author(s):  
Marilena Crupi ◽  
Rosanna Utano
Keyword(s):  
Betti Numbers ◽  
Upper Bounds ◽  
Exterior Algebra

scholarly journals Upper bounds for Betti numbers of multigraded modules

2007 ◽  
Vol 316 (1) ◽  
pp. 453-458
Author(s):  
Amanda Beecher
Keyword(s):  
Betti Numbers ◽  
Upper Bounds ◽  
Multigraded Modules

scholarly journals Upper Bounds on Betti Numbers of Tropical Prevarieties

2018 ◽  
Vol 4 (1) ◽  
pp. 127-136 ◽  
Author(s):  
Dima Grigoriev ◽  
Nicolai Vorobjov
Keyword(s):  
Betti Numbers ◽  
Upper Bounds

scholarly journals Certain homological invariants of bipartite kneser graphs

2021 ◽  
pp. 2250206
Author(s):  
Ajay Kumar ◽  
Pavinder Singh ◽  
Rohit Verma
Keyword(s):  
Lower Bound ◽  
Betti Numbers ◽  
Projective Dimension ◽  
Upper Bounds ◽  
Combinatorial Formula ◽  
Lower And Upper Bounds ◽  
Edge Ideals ◽  
Homological Invariants ◽  
Kneser Graphs ◽  
Combinatorial Data

In this paper, we obtain a combinatorial formula for computing the Betti numbers in the linear strand of edge ideals of bipartite Kneser graphs. We deduce lower and upper bounds for regularity of powers of edge ideals of these graphs in terms of associated combinatorial data and show that the lower bound is attained in some cases. Also, we obtain bounds on the projective dimension of edge ideals of these graphs in terms of combinatorial data.

scholarly journals Graded Betti Numbers of Balanced Simplicial Complexes

2021 ◽  
Author(s):  
Martina Juhnke-Kubitzke ◽  
Lorenzo Venturello
Keyword(s):  
Polynomial Ring ◽  
Betti Numbers ◽  
Simplicial Complexes ◽  
Upper Bounds ◽  
Combinatorial Type ◽  
Homogeneous Ideal ◽  
Graded Rings ◽  
Number Of Generators ◽  
Graded Betti Numbers ◽  
Linear Syzygies

AbstractWe prove upper bounds for the graded Betti numbers of Stanley-Reisner rings of balanced simplicial complexes. Along the way we show bounds for Cohen-Macaulay graded rings S/I, where S is a polynomial ring and $I\subseteq S$ I ⊆ S is a homogeneous ideal containing a certain number of generators in degree 2, including the squares of the variables. Using similar techniques we provide upper bounds for the number of linear syzygies for Stanley-Reisner rings of balanced normal pseudomanifolds. Moreover, we compute explicitly the graded Betti numbers of cross-polytopal stacked spheres, and show that they only depend on the dimension and the number of vertices, rather than also the combinatorial type.

scholarly journals Average Betti Numbers of Induced Subcomplexes in Triangulations of Manifolds

10.37236/8564 ◽  
2020 ◽  
Vol 27 (3) ◽  
Author(s):  
Giulia Codenotti ◽  
Jonathan Spreer ◽  
Francisco Santos
Keyword(s):  
Simplicial Complex ◽  
Lower Bounds ◽  
Commutative Algebra ◽  
Betti Numbers ◽  
Upper Bounds ◽  
Weighted Sums ◽  
Weighted Averages ◽  
Connected Sums ◽  
Graded Betti Numbers ◽  
Strongly Connected

We study a variation of Bagchi and Datta's $\sigma$-vector of a simplicial complex $C$, whose entries are defined as weighted averages of Betti numbers of induced subcomplexes of $C$. We show that these invariants satisfy an Alexander-Dehn-Sommerville type identity, and behave nicely under natural operations on triangulated manifolds and spheres such as connected sums and bistellar flips. In the language of commutative algebra, the invariants are weighted sums of graded Betti numbers of the Stanley-Reisner ring of $C$. This interpretation implies, by a result of Adiprasito, that the Billera-Lee sphere maximizes these invariants among triangulated spheres with a given $f$-vector. For the first entry of $\sigma$, we extend this bound to the class of strongly connected pure complexes. As an application, we show how upper bounds on $\sigma$ can be used to obtain lower bounds on the $f$-vector of triangulated $4$-manifolds with transitive symmetry on vertices and prescribed vector of Betti numbers.

Betti Numbers and Flat Dimensions of Local Cohomology Modules

2015 ◽  
Vol 58 (3) ◽  
pp. 664-672 ◽  
Author(s):  
Alireza Vahidi
Keyword(s):  
Upper Bound ◽  
Noetherian Ring ◽  
Local Cohomology ◽  
Betti Numbers ◽  
Upper Bounds ◽  
Commutative Noetherian Ring ◽  
Flat Dimension ◽  
Local Cohomology Modules ◽  
Cohomology Modules

AbstractAssume that R is a commutative Noetherian ring with non-zero identity, 𝔞 is an ideal of R, and X is an R-module. In this paper, we first study the finiteness of Betti numbers of local cohomology modules . Then we give some inequalities between the Betti numbers of X and those of its local cohomology modules. Finally, we present many upper bounds for the flat dimension of X in terms of the flat dimensions of its local cohomology modules and an upper bound for the flat dimension of in terms of the flat dimensions of the modules , and that of X.

scholarly journals Betti Numbers of Cut Ideals of Trees

2013 ◽  
Vol 4 (1) ◽  
Author(s):  
Samu Potka ◽  
Camilo Sarmiento
Keyword(s):  
Betti Numbers ◽  
Upper Bounds ◽  
Free Resolutions ◽  
Combinatorial Topology ◽  
Induced Subgraphs ◽  
Tree Graphs ◽  
Minimal Free Resolutions

Cut ideals, introduced by Sturmfels and Sullivant, are used in phylogenetics and algebraicstatistics. We study the minimal free resolutions of cut ideals of tree graphs. By employingbasic methods from combinatorial topology, we obtain upper bounds for the Betti numbers of thistype of ideals. These take the form of simple formulas on the number of vertices, which arise fromthe enumeration of induced subgraphs of certain incomparability graphs associated to the edgesets of trees.

scholarly journals Effective Cylindrical Cell Decompositions for Restricted Sub-Pfaffian Sets

2020 ◽  
Author(s):  
Gal Binyamini ◽  
Nicolai Vorobjov
Keyword(s):  
Betti Numbers ◽  
Cell Decomposition ◽  
Upper Bounds ◽  
Geometric Complexity ◽  
Cylindrical Cell ◽  
Cell Decompositions ◽  
Minimal Structure ◽  
Natural Measure ◽  
Definition Of

Abstract The o-minimal structure generated by the restricted Pfaffian functions, known as restricted sub-Pfaffian sets, admits a natural measure of complexity in terms of a format  ${{\mathcal{F}}}$, recording information like the number of variables and quantifiers involved in the definition of the set, and a degree  $D$, recording the degrees of the equations involved. Khovanskii and later Gabrielov and Vorobjov have established many effective estimates for the geometric complexity of sub-Pfaffian sets in terms of these parameters. It is often important in applications that these estimates are polynomial in $D$. Despite much research done in this area, it is still not known whether cell decomposition, the foundational operation of o-minimal geometry, preserves polynomial dependence on $D$. We slightly modify the usual notions of format and degree and prove that with these revised notions, this does in fact hold. As one consequence, we also obtain the first polynomial (in $D$) upper bounds for the sum of Betti numbers of sets defined using quantified formulas in the restricted sub-Pfaffian structure.

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