Freeness for 13 lines arrangements is combinatorial

Dimca, Alexandru; Ibadula, Denis; Măcinic, Anca

doi:10.1016/j.disc.2019.05.016

Cited by 12 publications

(13 citation statements)

References 16 publications

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“…Recently, Dimca, Ibadula, and Macinic confirmed Terao's conjecture for arrangements in C 3 with up to 13 hyperplanes [DIM19]. In joint work with Behrends, Jefferson, and Leuner, we confirmed Terao's conjecture for rank 3 arrangements with exactly 14 hyperplanes in arbitrary characteristic [BBJ + 21].…”

Section: Introductionsupporting

confidence: 63%

Computing the nonfree locus of the moduli space of arrangements and Terao's freeness conjecture

Barakat¹,

Kühne²

2021

Preprint

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In this paper, we show how to compute using Fitting ideals the nonfree locus of the moduli space of arrangements of a rank 3 simple matroid, i.e., the subset of all points of the moduli space which parametrize nonfree arrangements. Our approach relies on the so-called Ziegler restriction and Yoshinaga's freeness criterion for multiarrangements. We use these computations to verify Terao's freeness conjecture for rank 3 central arrangements with up to 14 hyperplanes in any characteristic.

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Section: Introductionsupporting

confidence: 63%

Computing the nonfree locus of the moduli space of arrangements and Terao's freeness conjecture

Barakat¹,

Kühne²

2021

Preprint

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“…[FV14,ACKN16]). Recently, Dimca, Ibadula, and Macinic confirmed Terao's conjecture for arrangements in \BbbC 3 with up to 13 hyperplanes [DIM19].…”

mentioning

confidence: 77%

On the Generation of Rank 3 Simple Matroids with an Application to Terao's Freeness Conjecture

Barakat¹,

Behrends²,

Jefferson³

et al. 2021

SIAM J. Discrete Math.

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\bfO \bfN \bfT \bfH \bfE \bfG \bfE \bfN \bfE \bfR \bfA \bfT \bfI \bfO \bfN \bfO \bfF \bfR \bfA \bfN \bfK \bfthree \bfS \bfI \bfM \bfP \bfL \bfE \bfM \bfA \bfT \bfR \bfO \bfI \bfD \bfS \bfW \bfI \bfT \bfH \bfA \bfN \bfA \bfP \bfP \bfL \bfI \bfC \bfA \bfT \bfI \bfO \bfN \bfT \bfO \bfT \bfE \bfR \bfA \bfO '\bfS \bfF \bfR \bfE \bfE \bfN \bfE \bfS \bfS \bfC \bfO \bfN \bfJ \bfE \bfC \bfT \bfU \bfR \bfE \ast MOHAMED BARAKAT \dagger , REIMER BEHRENDS \ddagger , CHRISTOPHER JEFFERSON \S , LUKAS K \" UHNE \P , \mathrm{\mathrm{\mathrm{ MARTIN LEUNER \| \bfA \bfb \bfs \bft \bfr \bfa \bfc \bft . In this paper we describe a parallel algorithm for generating all nonisomorphic rank 3 simple matroids with a given multiplicity vector. We apply our implementation in the high performance computing version of GAP to generate all rank 3 simple matroids with at most 14 atoms and an integrally splitting characteristic polynomial. We have stored the resulting matroids alongside with various useful invariants in a publicly available, ArangoDB-powered database. As a byproduct we show that the smallest divisionally free rank 3 arrangement which is not inductively free has 14 hyperplanes and exists in all characteristics distinct from 2 and 5. Another database query proves that Terao's freeness conjecture is true for rank 3 arrangements with 14 hyperplanes in any characteristic.\bfK \bfe \bfy \bfw \bfo \bfr \bfd \bfs . rank 3 simple matroids, integrally splitting characteristic polynomial, Terao's freeness conjecture, recursive iterator, tree-iterator, leaf-iterator, iterator of leaves of rooted tree, priority queue, parallel evaluation of recursive iterator, noSQL database, ArangoDB \bfA \bfM \bfS \bfs \bfu \bfb \bfj \bfe \bfc \bft \bfc \bfl \bfa \bfs \bfs \bfi fi\bfc \bfa \bft \bfi \bfo \bfn \bfs . 05B35, 52C35, 32S22, 68R05, 68W10 \bfD \bfO \bfI . 10.1137/19M1296744 \bfone . \bfI \bfn \bft \bfr \bfo \bfd \bfu \bfc \bft \bfi \bfo \bfn . In computational mathematics one often encounters the problem of scanning (finite but) large sets of certain objects. Here are two typical scenarios:\bullet Searching for a counterexample of an open conjecture among these objects.\bullet Building a database of such objects with some of their invariants. A database is particularly useful when the questions asked are relational, i.e., involve more than one object (cf. Remark 2.11). Recognized patterns and questions which a database answers affirmatively may lead to working hypotheses or even proofs by inspection (cf. Theorem 1.3).In any such scenario there is no need to simultaneously hold the entire set in RAM. It is hence important to quickly iterate over such sets in a memory efficient way rather than to enumerate them.The central idea is to represent each such set T as the set of leaves of a rooted tree T \bullet (cf. Appendix A). In other words, we embed T as the set of leaves in the bigger set of vertices V (T \bullet ). We then say that T \bullet classifies T . The internal vertices of the

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“…Since b 0 2 (B) = 5 • 7 + 1 = 36, this implies that B is nearly free for e 1 e 2 = 35 and free for e 1 e 2 = 36. However, no free arrangements can appear in the lattice isomorphism class of the nearly free A, because freeness is combinatorial for 13 lines arrangements, by [11].…”

Section: Then the Freeness Of A Depends Only On L(a)mentioning

confidence: 99%