Chordality, d-collapsibility, and componentwise linear ideals

Bigdeli, Mina; Faridi, Sara

doi:10.1016/j.jcta.2019.105204

Cited by 10 publications

(3 citation statements)

References 34 publications

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“…Recently, Bigdeli and Faridi gave a connection between the d-collapsibility and the chordal complexes; and proved that d-collapsibility is equivalent to the chordality of the Stanley-Reisner complexes of certain ideals [11]. For applications regarding Helly-type theorems, see [9,29,30].…”

Section: Introductionmentioning

confidence: 99%

On Vietoris--Rips complexes (with scale 3) of hypercube graphs

Shukla¹

2022

Preprint

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For a metric space (X, d) and a scale parameter r ≥ 0, the Vitoris-Rips complex VR(X; r) is a simplicial complex on vertex set X, where a finite set σ ⊆ X is a simplex if and only if diameter of σ is at most r. For n ≥ 1, let In denotes the n-dimensional hypercube graph. In this paper, we show that VR(In; 3) has non trivial reduced homology only in dimensions 4 and 7. Therefore, we answer a question posed by Adamaszek and Adams recently.A (finite) simplicial complex ∆ is d-collapsible if it can be reduced to the void complex by repeatedly removing a face of size at most d that is contained in a unique maximal face of ∆. The collapsibility number of ∆ is the minimum integer d such that ∆ is d-collapsible. We show that the collapsibility number of VR(In; r) is 2 r for r ∈ {2, 3}.

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Section: Introductionmentioning

confidence: 99%

On Vietoris--Rips complexes (with scale 3) of hypercube graphs

Shukla¹

2022

Preprint

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“…This class has been studied in [8]. Later, in [7] a modification of this variation of chordality was studied for simplicial complexes. Trying to compare the properties of chordal clutters and simplicial clutters, it turns out that, like chordal clutters, the ideal associated to a simplicial clutter has linear resolution over all fields and all of its graded Betti numbers can be identified explicitly (Corollary 3.1).…”

Section: Introductionmentioning

confidence: 99%

Multigraded minimal free resolutions of simplicial subclutters

Bigdeli¹,

Pour²

2020

Preprint

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This paper concerns the study of a class of clutters called simplicial subclutters. Given a clutter C and its simplicial subclutter D, we compare some algebraic properties and invariants of the ideals I, J associated to these two clutters, respectively. We give a formula for computing the (multi)graded Betti numbers of J in terms of those of I and some combinatorial data about D. As a result, we see that if C admits a simplicial subclutter, then there exists a monomial u / ∈ I such that the (multi)graded Betti numbers of I + (u) can be computed through those of I. It is proved that the Betti sequence of any graded ideal with linear resolution is the Betti sequence of an ideal associated to a simplicial subclutter of the complete clutter. These ideals turn out to have linear quotients. However, they do not form all the equigenerated square-free monomial ideals with linear quotients. If C admits ∅ as a simplicial subclutter, then I has linear resolution over all fields. Examples show that the converse is not true.

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“…The above definition was introduced by Hoefel and Mermin in [25] Herzog and Hibi [27] showed that all Gotzmann ideals are componentwise linear, and Bigdeli and Faridi [11] established a connection between square-free Gotzmann and Stanley-Reisner ideals of chordal complexes -a large class of componentwise linear ideals which can be defined via simplicial collapses. Open questions that arise from this are the following: Question 2.13.…”

Section: Introductionmentioning

confidence: 99%

Intersections of Amoebas

Juhnke-Kubitzke

Wolff²

2020

Discrete Mathematics &Amp; Theoretical Computer Science

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International audience Amoebas are projections of complex algebraic varieties in the algebraic torus under a Log-absolute value map, which have connections to various mathematical subjects. While amoebas of hypersurfaces have been inten- sively studied during the last years, the non-hypersurface case is barely understood so far. We investigate intersections of amoebas of n hypersurfaces in (C∗)n, which are genuine supersets of amoebas given by non-hypersurface vari- eties. Our main results are amoeba analogs of Bernstein's Theorem and Be ́zout's Theorem providing an upper bound for the number of connected components of such intersections. Moreover, we show that the order map for hypersur- face amoebas can be generalized in a natural way to intersections of amoebas. We show that, analogous to the case of amoebas of hypersurfaces, the restriction of this generalized order map to a single connected component is still 1-to-1.

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Chordality, d-collapsibility, and componentwise linear ideals

Cited by 10 publications

References 34 publications

On Vietoris--Rips complexes (with scale 3) of hypercube graphs

On Vietoris--Rips complexes (with scale 3) of hypercube graphs

Multigraded minimal free resolutions of simplicial subclutters

Intersections of Amoebas

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