Oriented : Applications - Chapters 6, 7 and 8

Kalle Klimkewitz 10.12.2013

Contents

1 Introduction 1

2 Catalog 1 2.1 Organisation ...... 2 2.2 Properties ...... 4 2.3 Generation ...... 6

3 Point Configurations 7 3.1 Basics ...... 7 3.2 A Conjecture Related to the Sylvester-Gallai Theorem ...... 9

4 Hyperplane Arrangements 9 4.1 Basics ...... 10

1 Introduction

In the presentation and this final report we want to talk about the applications of what we have learned so far in the doctoral thesis of L.Finschi [1]. This covers most of chapters 6, 7 and 8 of the thesis. We take a look at how we create a catalog of oriented matroids in section 2. Next, in section 3, we want to find out how to use this catalog to create a catalog of all possible point configurations. In section 4 we do the same thing for hyperplane arrangements. Note that whenever we specify numbers in definitons, theorems, etc. we refer to the number in the thesis.

2 Catalog

We want to create a catalog of all isomorphism classes of oriented matroids. To do this we want to use chapters 3 (Generation of Oriented Matroids and

1 Isomorphism Classes), 4 (Tope Graphs and Single Element Extensions) and 5 (Cocircuit Graphs and Single Element Extensions) of the thesis. That means we want to create all isomorphism classes inductively by sin- gle element extensions starting from the easy base cases when the number of elements equals the rank of the or when the rank is 2. In both cases there exists exactly one isomorphism class of oriented matroids. Note that we can always restrict to simple oriented matroids, as there exists a simple oriented matroid in every isomorphism class. In subsection 2.1 we will specify the way of organizing the calatog, in 2.2 we will investigate some the the properties this induces. In 2.3 we show how the catalog is generated.

2.1 Organisation First, one has to decide on how to organize this catalog. For this one has to decide on two basic questions: • how do we save the isomorphism classes, i.e. we have to decided on a data format • how do we order the classes, i.e. decide on an ordering principle Lets look at the ordering principle first. When deciding how to order the isomorphism classes we should try to order them by invariants of the class as otherwise the ordering might depend on the representative we choose for every class. These invariants include: the number of elements n, the rank r, the big face lattice, the tope graph and the cocircuit graph. Notice that the latter three do not directly induce an ordering, meaning if we are for example given two tope graphs its not clear which one is bigger. This is why we choose n and r to order by first, leading to the definiton of IC(n, r), a list that includes all isomorphism classes of oriented matroids with n elements of rank r. We thus only need to decide how to order inside IC(n, r). As there does not seem to be a canonical way to do so, this gives us the freedom to choose it somewhat arbitrarily such that it will give us a natural and practical overall ordering. Before we go into further detail we need to specify the data format that is used. For this we need some preliminaries from Chapter 0. With any basis of the underlying matroid and every element in it, we can associate a cocircuit that is well defined, called the fundamental cocircuit: Definiton (0.9.3, Fundamental Cocircuit). For an oriented matroid M, a basis 0 B and an e ∈ B we call the cocircuit X determined by B\e ⊆ X and Xe = + the fundamental cocircuit of B w.r.t. e This cocircuit can be used to define a basis orientation, which acts as a sign variant of the : Definiton (0.9.4, Basis Orientation). For an oriented matroid M and its or- dered bases (B) a map χ :(B) → {−, +} is called a basis orientation if

2 • χ is alternating, i.e. if for any ordered basis (B) and permutation π we have χ(π(B)) = sign(π)χ(B). • for all (B) ∈ (B), e ∈ B, f 6∈ B s.t B\e ∪ f ∈ B we have χ(B : e → f) = XeXf χ(B) where X is the fundamental cocircuit of w.r.t. B and e (i.e. we can calculate the value of the new basis using the value of the old basis and the fundamental cocircuit). Such a basis orientation is unique up to negation, as stated by the next theorem. Theorem (0.9.5, Las Vergnas). Every oriented matroid has exactly two basis orientations χ and −χ. We extend this unique basis orientation to all r-subsets and get the chirotope which determines the oriented matroid: Definiton (0.9.6, Chirotope). Let M be oriented matroid of rank r on n ele- ments. We call {χ, −χ} the chirotope of M if χ is a map defined on all ordered r-subsets of E s.t. • restricted to the bases, χ is a basis orientation • χ = 0 for all non-basis r-subsets Proposition (0.9.7). The chirotope together with rank r and E determines M.

The proposition is easily proven by getting the cocircuits from the bases, from which we get all covectors in M. We use the chirotope to save the matroid. For every matroid we need to save n one sign for every r-subset of {1, . . . , n}, thus r signs in total. This number is the same for every matroid in IC(n, r). We order the subsets in reverse lexicographic ordering. That means that we order the elements in every subset increasing and order the subsets by the largest element first, than by the second largest and so forth. Lets look at an example, the ordering of 2-subsets of 4 elements: Example. The usual ordering would be: 12,13,14,23,24,34 (ordering by the lowest element first) whereas we take 12,13,23,14,24,34

3 Note that this give us the following two interesting properties. The first 2 sets do not contain 4 and are exactly the 2-subsets of 3 elements in the reverse 3 lexicographic ordering. The last 1 sets on the other hand contain 4 and if we delete the 4 from every set, we are left with the 1-subsets of 3 elements, also n−1 in reverse lexicographic ordering. This generalizes to the first r signs and n−1 the last r−1 signs for r-subsets of n elements, which will be used in the later proofs. To save the matroid we now always take χ or −χ s.t. first non-zero entry is +. More formally:

3 Definiton (Encoding). M oriented matroid on E = {1, . . . , n} of rank r. De- n note χ(M) ∈ {−, +, 0}(r) the encoding of M by the lexicographically positive map in the chirotope of M where the signs are given according to the reverse lexicographically sorted r-subsets. Now we can give an answer to the Isomorphism Class Representation Prob- lem posed in Chapter 3. For every class of isomorphic oriented matroids we choose the representative s.t. χ is lexicographically maximal, where − < + < 0. To find the representative of given simple oriented matroid we might need to consider all permutations and reorientations. Again, more formally and to introduce the notation: Definiton (Representative). M oriented matroid on E = {1, . . . , n}. Denote by rep(M) the uniquely determined simple oriented matroid on E which is in the same isomorphism class and for which χ(rep(M)) is lexicographically maximal. With this we can now specify how we want to order inside IC(n, r): we order the classes such that the encoding of the representatives is lexicographically increasing. Denote by IC(n, r, c) the isomorphism class that comes at the cth position in this list. This completes the specification of the organization of the catalog.

2.2 Properties Here we want to investigate three properties of the catalog. The first two will be used in the generation process, the third is a useful property about uniform oriented matroids. Lemma (6.3.1). M simple oriented matroid on E = {1, . . . , n} of rank r. Then

n−1 1. If n ∈ E is not a coloop then the first r signs of χ(M) are not all 0 and are equal to the signs of χ(M\n) (in the same order)

n−1 2. If n ∈ E is a coloop then the first r signs of χ(M) are all 0 and the n−1 last r−1 signs of χ(M) are equal to the sign of χ(M\n) (in the same order)

Proof. To prove this we use reverse lexicographic ordering and the properties that were mentioned earlier. Note that n is a coloop if and only if it is part of every basis. Thus, if n is not a coloop there exists an r-subset that does not include n n−1 (and corresponds to one of the first r signs) and that is a basis meaning it has a non-zero entry. Notice that the rank of M is equal to the rank of M\n and every basis of the latter is also a basis of M. Restricting χ to the deletion minor leads to a basis orientation of the deletion minor (the axioms are fulfilled), which proves the first part. If n is a coloop however all r-subsets without n are not bases, such that n−1 the first r signs are 0. Note that the rank decreases by 1 if n is deleted.

4 ⇒

Figure 1: Example with IC(6, 4) of the first part of Lemma 6.3.1.

⇒

Figure 2: Example with IC(6, 4) of the second part of Lemma 6.3.1.

Restricting χ to the bases of M\n again satisfies the properties of the basis orientation proving the second part. For an intuitively accessible example look at figure 1 (for the first part) and at figure 2 (for the second part). Note that this would work for every oriented matroid, not only the repre- sentative of its class. Next we look at the representative of each class. Proposition (6.3.2). M simple oriented matroid on E = {1, . . . , n} of rank r, s.t. rep(M) = M 1. rep(M\n) = M\n 2. M has a coloop if and only if n is a coloop. 3. If n is not a coloop then χ(rep(M\n)) = χ(M\n) is lexicographically maximal among all χ(rep(M\i)), i ∈ {1, . . . , n}.

5 The proof is a little lengthy but not difficult. It is proven by contradiction using the previous lemma. Lastly we look uniform oriented matroids and the order inside IC(n, r). Corollary (6.3.3). M simple oriented matroid on E = {1, . . . , n} of rank r, s.t. rep(M) = M

1. The first sign of χ(M) is nonzero (thus +) ⇔ M is uniform. 2. All isomorphism classes of uniform oriented matroids come at the begin- ning. 3. All isomorphism classes of oriented matroidswith coloops come at the end.

Proof. Remember that an oriented matroid of rank r is uniform if and only if all r-subsets are bases, i.e. if the encoding contains no 0. As we chose the representative lexicographically maximal, this happens if and only if the first entry is non-zero. The two other points are simple due to the fact that we order the classes lexicographically increasing and using Lemma 6.3.1.

2.3 Generation To generate IC(n, r) we use the following procedure, assuming we have already generated IC(n − 1, r) and IC(n − 1, r − 1). • Initialize IC(n, r) = ∅

• For every c from 1 to |IC(n − 1, r)|do :

– Let M be representative of IC(n − 1, r, c). – Compute all simple single element extensions M0, where new element is not a coloop (using LocalizationsPatternBacktrack from Chapter 5) – Compute the representatives n−1 – Keep those where first r signs are equal too χ – Sort remaining extensions w.r.t to lexicographic ordering – Remove multiple entries – Append to IC(n, r).

• For every c from 1 to |IC(n − 1, r − 1)|do : – Let M be representative of IC(n − 1, r − 1, c) n−1 – Append r 0’s to χ in the beginning – Append to IC(n, r)

6 There are a couple of things to note here. First, notice that we do not compare or sort extensions coming from different deletion minors. They do not produce the same class of oriented matroids by the way the catalog is set up. Second, we see that uniform matroids are only generated from uniform matroids for which there exists more efficient methods that can be used as a subroutine. Third, we see that we have to compute the representative of a lot of oriented matroids which is, as mentioned earlier, computationally hard and there does not seem to be a way of speeding up this process. The next proposition tells us that this procedure actually works. It basically is the induction step in proving that all classes are generated correctly. The base cases are treated in earlier chapters and are, as also mentioned above, the cases when n = r and r = 2. Proposition (6.4.1). If IC(n−1, r−1) and IC(n−1, r) are generated correctly, this procedure will generate IC(n, r) correctly. The proof is once again somewhat lengthy. There are three things that we have to show. We need to show that every class is generated exactly once, that it is represented correctly and that it is at the correct position. As stated above, this holds even though we do not compare and sort extensions from different matroids. To prove this we use lemma 6.3.1 and proposition 6.3.2. The resulting catalog of oriented matroids can be found online at [3].

3 Point Configurations

This catalog of oriented matroids can be used for generating a complete listing of point configurations, including degenerate ones.

3.1 Basics

Lets look a finite set of points P = {v1, . . . , vn} in Rd. Every hyperplane given by its normal vector x and a translation xd+1 gives a sign for each point, which is + if the point lies on the same side of the hyperplane as its normal vector. More formally the covector X defined by P and a hyperplane h is given by Pd e Xe = sign( i=1 vi xi + xd+1). We want to force this to be a scalar product. e This is done by lifting the points one dimension higher and setting vd+1 = 1. By setting the columns of a matrix A as ve, we then get the set of covectors induced by P as F(P) = {sign(Atx)|x ∈ Rd+1}. This is always the set of covector from a realizable oriented matroid, as we saw in the first chapters of the thesis. An example with five points in the plane can be found in figure 3. Notice that we can always translate an hyperplane such that all points lie on the plus-side of the hyperplane. In other words, the set of covectors always includes the covector which is all +, leading to the following definition: Definiton (Acyclic Oriented Matroid). An oriented matroid where there exists an X ∈ F with Xe = + for all e is called acyclic.

7 Figure 3: Five points in the plane and a hyperplane with the corresponding covector

Figure 4: A 3d sphere arrangement corresponding to a 2d point configuration

We are now interested in finding all possible point configurations, or more precisely the relabeling classes of F(P): Definiton (Order Type). For a point configuration P we call the relabeling class of F(P) the order type of P.

Notice that the order types of point configurations correspond one-to-one to the relabeling classes of realizable acyclic oriented matroids. We saw above how we get from a point configuration to a realizable acyclic oriented matroid. For the other direction assume we are given a realizable acyclic oriented matroid which can be given as a central sphere arrangement in dimension r. As it acyclic there exists a cell corresponding to the all-positive covector. By turning the the sphere s.t. the vector (0, 0,..., 0, 1) is contained in this cell we can be sure that all normal vectors of the sphere-inducing hyperplanes lie in the upper half of the space. We can scale them s.t. they all have last component 1 giving us a point configuration in the d-dimensional space which has last component 1. For an example of a sphere arrangement in 3 dimensions giving rise to a 2-dimensional point configuration see figure 4. Checking if a given matroid is realizable is in general NP-hard. We thus generalize the notion of order types to abstract order types. Definiton (Abstract Order Type). We call the relabeling class of an acyclic oriented matroid an abstract order type.

8 We can generate abstract order types from the oriented matroid catalog. For this we need to consider all possible reorientations of all isomorphism classes of a given IC(n, r). This will give us a list of abstract order types. To get order types is then NP-hard in general. Deciding realizability corresponds to solving a polynomial program as we get constraints on the sub- of a matrix. We know however that whenever d = 2 and n ≤ 8 all matroids are realizable.The results can be found on [3] again, the corresponding realizations in [1]. Note that uniform oriented matroids correspond to non-degenerate point configurations.

3.2 A Conjecture Related to the Sylvester-Gallai Theo- rem Here we want to investigate a conjecture related to the Sylvester-Gallai Theorem which was disproven using the catalog of abstract order types. Lets look at the Sylvester-Gallai theorem first:

Theorem (Sylvester-Gallai). For every non-collinear point configuration in R2 There exists a line that contains exactly two points. There are many different proofs, some of which are quite easy. The con- jecture introduces an additional requirement on the line containing exactly two points. Conjecture (da Silva and Fukuda). For every non-collinear point configuration in R2 and each separating line through no points, for which the number of points on the left and on the right differs by at most 1, there exists a line through exactly one point on the left and exactly one on the right. If we translate this to simple, acyclic, realizable oriented matroids a line through no points becomes a tope (as topes in simple oriented matroids are covectors without zeros). A tope X is balanced if and only if ||X+| − |X−|| ≤ 1. Dropping the realizability constraint we get a slightly stronger conjecture:

Conjecture (Oriented matroid version of the da Silva and Fukuda Conjec- ture). Let M be simple acyclic oriented matroid of rank 3. For every bal- anced tope X there exists a pair {e, f} ∈ E of elements s.t. Xe = −Xf 6= 0 and spanM({e, f}) = {e, f} or, equivalently, there exists a cocircuit Y s.t. Y 0 = {e, f}.

This conjecture was checked against the catalog of abstract order types and holds true for n ≤ 8 and n = 10. For n = 9 however a counterexample was found, which was found to be realizable and can be seen in figure 5. This counterexample thus disproves both versions of the conjecture.

4 Hyperplane Arrangements

Instead of point configurations we can also consider hyperplane arrangements.

9 Figure 5: A counterexample to the Conjecture of da Silva and Fukuda

4.1 Basics

So let Q = {h1, . . . , hn} in Rd be a finite set of hyperplanes. Every he is de- e d e scribed by normal vector v ∈ R and translation vd+1 ∈ R. Similar to before Pd e the covector X defined by Q and point x can be given by Xe = sign( i=1 vi xi + vd+1). Here we cannot simply set xd+1 as we did for point configurations. In- stead we make this scalar product by forcing xd+1 = 1 which we do by introduce g g a new hyperplane with vi = 0 for all i ∈ {1, . . . d} and vd+1 = 1 (called the infinity element), which we can basically use to check if xd+1 = 1 can be fulfilled by scaling x appropriately. Intuitively the infinity element takes on the role of the all-positive cell in point configurations: it tells us how to turn a given sphere arrangement to get the hyperplane arrangement. By setting the columns of ma- trix A as ve we again get the set of covectors as F(Q) = {sign(Atx)|x ∈ Rd+1}. We can use the above to get a catalog of Abstract Dissection Types (equiv- alence classes of oriented matroids with a marked infinity element with respect to isomorphisms that map the infinity element to the infinity element) from ori- ented matroid catalog by marking each element as infinity element once. Similar to order types for point configuration this will give us a list of all hyperplane arrangements. For n hyperplanes in Rd consider IC(n + 1, d + 1). As before the resulting catalog can be found on [3], the correspond realizations in [1].

References

[1] L. Finschi, A Graph Theoretial Approach for Reconstruction and Generation of Oriented Matroids, Dissertation (2001) [2] I.P.F. da Silva, K.Fukuda, Isolating points by lines in the plane Journal of (1998)

[3] L.Finschi, http://www.om.math.ethz.ch/

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