Toric chordality

Adiprasito, Karim

doi:10.1016/j.matpur.2017.05.017

Cited by 16 publications

(20 citation statements)

References 31 publications

(52 reference statements)

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“…The following recent result of Adiprasito [Adi15] generalizes the lower bound theorem, and will be crucial in our proof of Conjecture 1.1(i).…”

mentioning

confidence: 83%

“…In [ANS15] we provided a notion of higher chordality of simplicial complexes and showed that it generalizes the classical notion of chordal graphs. In [Adi15] the first named author introduced toric chordality, a powerful algebraic tool to study chordality in the stress-space of the simplicial complex as studied by Lee [Lee96]. He related this algebraic notion of chordality to the higher chordality notions of [ANS15] and derived, among many other results, a quantitative version of the GLBC in terms of the topological Betti numbers of induced subcomplexes.…”

Section: Introductionmentioning

confidence: 99%

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A Geometric Lower Bound Theorem

2016

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We resolve a conjecture of Kalai relating approximation theory of convex bodies by simplicial polytopes to the face numbers and primitive Betti numbers of these polytopes and their toric varieties. The proof uses higher notions of chordality.Further, for C 2 -convex bodies, asymptotically tight lower bounds on the g-numbers of the approximating polytopes are given, in terms of their Hausdorff distance from the convex body.

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“…The following recent result of Adiprasito [Adi15] generalizes the lower bound theorem, and will be crucial in our proof of Conjecture 1.1(i).…”

mentioning

confidence: 83%

Section: Introductionmentioning

confidence: 99%

A Geometric Lower Bound Theorem

2016

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“…This almost is a Lefschetz-type result that characterizes primitive Betti numbers, compare also [Sta96]. We refer to [Adi15] for related applications towards a quantitative lower bound theorem, and also Remark 5.18 for a small application. and further developed in [AB12].…”

Section: Corollary 425 Let ∆ Be a Simplicial Ball And Assume That ∂mentioning

confidence: 65%

Relative Stanley–Reisner theory and Upper Bound Theorems for Minkowski sums

Adiprasito

Sanyal

2016

Publ.math.IHES

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Abstract. In this paper we settle two long-standing questions regarding the combinatorial complexity of Minkowski sums of polytopes: We give a tight upper bound for the number of faces of a Minkowski sum, including a characterization of the case of equality. We similarly give a (tight) upper bound theorem for mixed facets of Minkowski sums. This has a wide range of applications and generalizes the classical Upper Bound Theorems of McMullen and Stanley.Our main observation is that within (relative) Stanley-Reisner theory, it is possible to encode topological as well as combinatorial/geometric restrictions in an algebraic setup. We illustrate the technology by providing several simplicial isoperimetric and reverse isoperimetric inequalities in addition to our treatment of Minkowski sums.The Upper Bound Theorem (UBT) for polytopes is one of the cornerstones of discrete geometry. The UBT gives precise bounds on the 'combinatorial complexity' of a convex polytope P as measured by the number of k-dimensional faces f k (P ) in terms of its dimension and the number of vertices. Upper Bound Theorem for polytopes. For a d-dimensional polytope P on n vertices and 0where Polytopes attaining the upper bound are called (simplicial) neighborly polytopes and are characterized by the fact that all non-faces are of dimension at least d 2 . Cyclic polytopes are a particularly interesting class of neighborly polytopes whose combinatorial structure allows for an elementary and explicit calculation of f k (Cyc d (n)) in terms of d and n; cf. [Zie95, Section 0]. The UBT was conjectured by Motzkin [Mot57] and proved by McMullen [McM70]. One of the salient features to note is that for given d and n there is a polytope that maximizes f k for all k simultaneously -a priori, this is not to be expected.In this paper we will address more general upper bound problems for polytopes and polytopal complexes. To state the main applications of the theory to be developed, recall that the Minkowski sum of polytopes and game theory. Our methods also apply to the study of mixed faces and we establish strong upper bounds and in particular characterize the case of equality in the most important case. Upper Bound Theorem for mixed facets. The number of mixed facets of a Minkowski sum is maximized by Minkowski neighborly families.From discrete geometry to combinatorial topology to commutative algebra. An intriguing feature of the UBT is that its validity extends beyond the realm of convex polytopes and into combinatorial topology. Let ∆ be a triangulation of the (d − 1)-sphere and, as before, let us write f k (∆) for the number of k-dimensional faces. For example, boundaries of simplicial d-dimensional polytopes yield simplicial spheres, but these are by far not all. The UBTM too will be the consequence of a statement in the topological domain that we derive using algebra, though we will also briefly comment on a geometric approach to the problem. The appropriate combinatorial/topological setup for the UBPM is that of relative simplicial complexes: A re...

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“…Here ∂∆ i denotes the boundary of an i-simplex and C k is the boundary of a k-gon. 1 In the following two lemmas we compute the τ -vectors of these two minimal cases.…”

Section: Nearly Stacked Spheresmentioning

confidence: 99%

“…Put differently, τ i is the weighted average of the i-th reduced Betti number of all induced subcomplexes C[W ], with respect to weights that are uniform on subsets W ⊆ V of equal size and add up to 1 |V |+1 for each size j ∈ {0, . .…”

Section: Introductionmentioning

confidence: 99%

Average Betti Numbers of Induced Subcomplexes in Triangulations of Manifolds

Codenotti

Spreer²,

Santos³

2020

Electron. J. Combin.

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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.

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Toric chordality

Cited by 16 publications

References 31 publications

A Geometric Lower Bound Theorem

A Geometric Lower Bound Theorem

Relative Stanley–Reisner theory and Upper Bound Theorems for Minkowski sums

Average Betti Numbers of Induced Subcomplexes in Triangulations of Manifolds

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