Abstract
We introduce a greedy algorithm optimizing arrangements of lines with respect to a property. We apply this algorithm to the case of simpliciality: it recovers all known simplicial arrangements of lines in a very short time and also produces a yet unknown simplicial arrangement with 35 lines. We compute a (certainly incomplete) database of combinatorially simplicial complex arrangements of hyperplanes with up to 50 lines. Surprisingly, it contains several examples whose matroids have an infinite space of realizations up to projectivities.
Highlights
A simplicial arrangement is a finite set of linear hyperplanes in a real vector space which decomposes its complement into open simplicial cones, cf. [17]
We find a real simplicial arrangement of rank three with 35 lines which does not appear in any previous table of simplicial arrangements
This is why the above algorithm will mostly find matroids with rational realizations and miss most of the interesting examples. To address this problem we just choose a subset F of the lines which should never be removed. This way the algorithm will regularly add lines generated by F, if F contains some “irrational” entries, they will remain all the time and it is more likely that these “irrationalities” are enforced by the resulting matroid structure: Algorithm 4.5 GreedyArrFiniteFieldAlgebraic (P, n, q, w, g) Greedy search for arrangements of hyperplanes over a finite field with given algebraic elements
Summary
A simplicial arrangement is a finite set of linear hyperplanes in a real vector space which decomposes its complement into open simplicial cones, cf. [17]. A classification of simplicial arrangements, even in the case of dimension three, has not been achieved in full generality yet. There is a topological result by Deligne [13] and there are some classifications of smaller classes, as in [9], [10], or [12]. No explicit approach to a classification is known. In this early stage of investigations, it is common to collect examples as in [2,6,15,16]
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