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11 Greedy search for arrangements

This is an implementation of a basic greedy algorithm to construct arrangements satisfying a certain property, based on [Cun22]. A variant was used in [MMR24] to construct an arrangement which is \(4\)-formal with a restriction which is not \(3\)-formal.

11.1 Constructions

11.1-1 HArrGreedySearch

Returns: A hyperplane arrangement

Constructs a greedy search for arrangements over GField with NumberOfHs hyperplanes in dimension dim.

• PropTargetFct should be a function taking a hyperplane arragement as argument, returning a non-negative real number, which the search tries to minimize, and such that value=0 means the arrangement satisfies the desired property.

• MaxNoIterations is the maximal number of steps to be carried out in the search (one step = exchanging a hyperplane).

gap> HArrGreedySearch(13,3,GF(17),CharPolySplits,400);
GreedySearch over GF(17) for arrangements:
 - of rank 3
 - with 13 hyperplanes.

11.1-2 CharPolySplits

A target/score function for splitting of the characteristic polynomial of A over \(\mathbb{Z}\).

11.1-3 DistMSetInvL

A target/score function for comparing MSetInvL(B) and MSetInvL(B) (see MSetInvL (2.2-7)).

11.2 Attributes

11.2-1 GreedySearchRun

Returns: a function

Run the search algorithm of GS which will return an arrangement or fail.

gap> GS:=HArrGreedySearch(13,3,GF(17),CharPolySplits,400);
GreedySearch over GF(17) for arrangements:
 - of rank 3
 - with 13 hyperplanes.
gap> A:=GreedySearchRun(GS)();
227 Iterations - <HyperplaneArrangement: 13 hyperplanes in 3-space>
gap> ExpArr(A);
[ 1, 6, 6 ]
gap> HArrIsSupersolvable(A);
true
gap> A:=GreedySearchRun(GS)();
358 Iterations - <HyperplaneArrangement: 13 hyperplanes in 3-space>
gap> ExpArr(A);
[ 1, 6, 6 ]
gap> HArrIsSupersolvable(A);
false
gap> HArrIsFree(A);
true

11.2-2 GreedySearchGF

Returns: a field

The Galois field over which to search for arrangements.

11.2-3 GreedySearchDimension

11.2-4 GreedySearchNOfHs

11.2-5 GreedySearchTargetFct

11.2-6 GreedySearchMaxIterations

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