Retracts and algebraic properties of cut algebras

Römer, Tim; Madani, Sara Saeedi

doi:10.1016/j.ejc.2017.11.002

Cited by 7 publications

(16 citation statements)

References 28 publications

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“…In our particular case of a monomial cut ideal I, the fiber ring F (I) is just the cut algebra of I which was introduced by Sturmfels and Sullivant [21] and has been further studied, in particular, by Römer-Saeedi Madani [16,17] and Koley-Römer [13].…”

Section: Freiman Monomial Cut Idealsmentioning

confidence: 99%

“…Sturmfels and Sullivant [21] started an interesting connection to algebraic geometry and commutative algebra by introducing toric cut ideals and -algebras, which have been intensively studied in the last decade. See, e.g., [7,14,16,17,18,19,20].…”

Section: Introductionmentioning

confidence: 99%

“…Thus, it suffices to show that I(P 3 ) and I(K 2 ⊔ K 3 ) are Freiman. This can easily be checked by CoCoA [3], by using the fact that the analytic spread of I(G) is equal to E(G)| + 1, as shown in [16,Formula (2)].…”

mentioning

confidence: 99%

“…Conversely, if I(G) is Freiman, then Theorem 6.1 together with [16, Proposition 3.1] and [16,Theorem 6.10] imply that G must be one of the graphs in the list. Here we use again that I(K 2 ⊔ K 2 ) = I(P 3 ), I(K 2 ⊔ K 3 ) = I(K 2 # K 1 K 3 ), and that I(K 4 ), which appears in the statement of [16,Theorem 6.10], has 4-linear resolution.…”

mentioning

confidence: 99%

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Classes of cut ideals and their Betti numbers

Herzog¹,

Rahimbeigi²,

Römer³

2021

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We study monomial cut ideals associated to a graph G, which are a monomial analogue of toric cut ideals as introduced by Sturmfels and Sullivant. Primary decompositions, projective dimensions, and Castelnuovo-Mumford regularities are investigated if the graph can be decomposed as 0-clique sums and disjoint union of subgraphs. The total Betti numbers of a cycle are computed. Moreover, we classify all Freiman ideals among monomial cut ideals.

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Section: Freiman Monomial Cut Idealsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Classes of cut ideals and their Betti numbers

Herzog¹,

Rahimbeigi²,

Römer³

2021

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“…In the particular case of cut polytopes, the aforementioned toric algebras and their defining ideals, which were studied first in [34], are called cut algebras and cut ideals, respectively. For further studies around cut algebras and ideals, see, e.g., [15,22,23,26,27,32,33]. For applications to algebraic statistics related to binary graph models, Markov random fields and phylogenetic models on split systems as a generalization of binary Jukes-Cantor models, see for example [34].…”

Section: Introductionmentioning

confidence: 99%

Cycle algebras and polytopes of matroids

Römer,

Madani

2021

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Cycle polytopes of matroids have been introduced in combinatorial optimization as a generalization of important classes of polyhedral objects like cut polytopes and Eulerian subgraph polytopes associated to graphs. Here we start an algebraic and geometric investigation of these polytopes by studying their toric algebras, called cycle algebras, and their defining ideals. Several matroid operations are considered which determine faces of cycle polytopes that belong again to this class of polyhedral objects. As a key technique used in this paper, we study certain minors of given matroids which yield algebra retracts on the level of cycle algebras. In particular, that allows us to use a powerful algebraic machinery. As an application, we study highest possible degrees in minimal homogeneous systems of generators of defining ideals of cycle algebras as well as interesting cases of cut polytopes and Eulerian subgraph polytopes.

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