Excluded Minority of P 8 for GF ( 4 )-Representability

One of the mainly interesting things of matroid theory is the representability of a matroid. Finding the set of all excluded minors for the representability is the solution of the representability. In 2000, Geelen, Gerards and Kapoor proved that ( ) { } * 2,6 4,6 7 7 6 8 , , , , , U U F F P P − − is the complete set of GF(4)-representability. In this paper, we show that 8 P is an excluded minor for GF(4)-representability.


Introduction
Matroid theory dates from the 1930's when Whitney first used the term matroid in his basic paper [1].Matroid theory is a common generalization of linear independence of graphs and matrices.One of the mainly interesting things of matroid theory is the representability of a matroid.One problem of representability is to find the fields over which the given matroid is representable.The other problem is to find the excluded minors for which the matroid is representable over the given field.It was found the complete set of excluded minors for two or three element fields [2]- [5].In 1984, Kahn and Seymour conjectured that P is also excluded minor for GF(4)-representability in his brief note [6].
Geelen, Gerarads and Kapoor proved that it is enough to add 8 P to the list of Kahn and Seymour [7].It is not an easy problem to find the excluded minors for GF(q)representability when q is more than 4. Instead, we have the conjecture by Rota that the number of excluded minors are finite for any prime powers q [8].In this paper, we study the properties of minor and show that 8 P is excluded minor for GF(4)representability deliberately.

Matroid
Matroid theory has exactly the same relationship to linear algebra as does point set topology to the theory of real numbers.That is, point set topology postulate the properties of the open sets of real line and matroid axiomatize the character of the independent set in vector space: Definition 2.1.A matroid M is a finite set E and a collection  of subsets of E satisfying the following three conditions: By the definition, for a finite vector space V and for the collection of linearly independent subsets  of vectors of V, ( )  is a matroid, M is called a matroid on E. Also E which is denoted by ( ) ( ) is called the set of independent set in M. A subset of E that is not in  is called dependent set.On the other point of view, matroid can be defined by abstraction of the properties of the cycles of a graph.Let G be a graph and let ( ) E G be the set of all edges of G. Also let  be the set of all cycles in G. Then  has the following properties: C of  such that ( ) Now, let  be a subset of the power set 2 E of a finite set E. If  satisfies the conditions (C 1 ), (C 2 ) and (C 3 ), then  is called the set of circuits of a matroid on E. Let  be the set of circuits of a matroid M.Then, the set  of all subsets of E which contain no member of  satisfies the independent conditions (I 1 ), (I 2 ) and (I 3 ).Also for a matroid ( ) of all minimal dependent set M satisfies the three circuits conditions.Thus, the matroid defined by circuits is the same as the one defined by independent sets.We need two other definitions of a matroid.Definition 2.2.Let E be a finite set and r be a function from 2 E to the set of nonnegative integers and satisfies the following conditions: . Then, r is called the rank function of a matroid M on E. Let M be the matroid ( ) Then, it is easy to see that the pair ( ) , | X X  is a matroid.We call this matroid the restriction of M to X or the deletion of E X − from M. It is denoted by will be denoted by ( ) r M .On the other hand, if r is a rank function of a matroid M, the set is the set of independent set of M. Thus, the definition of a matroid by a rank function is equivalent to the definition by independent sets.One of the definitions of matroid is the one by closure operator.Throughout this thesis, by E means a finite set.
Definition 2.3.Let cl be a function from 2 E to 2 E satisfying the following; Then cl is called the closure operator of a matroid M on E. Let M be a matroid on E with the rank function r.Define cl to be the function from for all X E ⊂ .Then, we can see that cl satisfies (CL 1 )-(CL 4 ).On the other hand, if cl is the closure operator of a matroid M on E, then satisfies the independent axioms and ( )  is the matroid having closure operator cl.Thus, the four definitions of matroid defined by the above are equivalent.We can see that there are a lot of equivalent definition of matroid.We have to introduce one more definition of matroid.
For a matroid M, the maximal independent set in M is called a basis or base of M. It is easy to see that every basis element has the same cardinal number.Let x B B ∈ − then there is Conversely, let  be a collection of subsets of E satisfying the axioms (B 1 ) and (B 2 ).

{ }
| for some , E  is a matroid having  as its collection of bases.If M is a matroid and ( ) cl X the closure or span of X in M, and we write this as , then X is called a flat or a closed set of M. A hyperplane of M is a flat of rank ( ) . Let M be a matroid and ( ) x B B ∈ − , then there is an element . The matroid * M having the set of all basis element B * (M) is called the dual of M. Thus ( ) ( ) and ( ) . The bases of * M are called cobases of M. Similarly, the circuits, hyperplanes, independent sets and spanning sets of * M are called cocircuits, cohyperplanes, coindependent sets, and cospanning sets of M. The next result gives some elementary relationships between these sets.Proposition 2.4.Let M be a matroid on a set E and suppose X E ⊂ .Then 1) X is independent if and only if 3) X is a hyperplane if and only if E X − is cocircuit. 4)X is a circuit if and only if E X − is a cohyperplane.Proof. 1) Let X be an independent set in M.Then, X B ⊂ for some basis element B of M. Thus E B E X − ⊂ − and ( ) ( ) ) is the one obtained by applying 3) to Let's remind of the finite fields.If F is a finite field, then F has exactly p k -elements for some prime p and some positive integer k.Indeed, for all such p and k, there is a unique field ( ) k GF p having p k -elements.This field is called the Galois field of order p k .When coincides with p  , the ring of integers modulo p.When can be constructed as follows.Let ( ) h ω be a polynomial of degree k with coefficients in p  and suppose that the polynomial is irreducible.Consider the set S of all polynomials in ω that have degree at most , we can take the irreducible polynomal to Let A be a matrix over a field F.Then, the collection of independent column vectors  of A satisfies the independent axioms of matroid.Thus, ( ) M M ≅ , if there is a bijection M A for a matrix A over F, then M is called F-representable.In the sequel, by F we mean a finite field.
We call an element e a loop of a matroid M if { } e is a circuit of M.Moreover, if f and g are element of ( ) E M such that { } , f g is a circuit, then f and g are said to be parallel in M. A parallel class of M is a maximal subset X of ( ) E M such that any two distinct members X are parallel and no member of X is a loop.A parallel class is trivial if it contains just one element.If M has no loops and no non-trivial parallel classes, it is called a simple matroid or a combinatorial geometry.

Uniform Matroid U m,n
Let m and n be non-negative integers such that m n ≤ .Let E be a set of cardinality n and  be all subsets of E of cardinality less than or equal to m.This is a matroid on E, called the uniform matroid of rank m and denoted by n m U , .By definition, the set of basis ( ) , and the set of circuits ( )

Affine Matroid
Now, we are going to define affine matroid.A set { } , , , , , , k a a a  of F, not all zero, such that , , , is equivalent to each of the followings; is linearly dependent, where ( ) , , , , , . Let A be the ( ) M A is called an affine matroid over F.
In particular, if ( ) , and m E F = , then the affine matroid is denoted by ( ) , AG m q .In general, if M is an affine matroid over  of rank ( ) If three elements form a circuit, the corresponding points are collinear.Likewise, if four elements form a circuit, the corresponding points are coplanar.In such a diagram, the lines need not be straight and the planes may be twisted.Certain lines with fewer than three points on them will be marked as part of the indication of a plane, or as construction lines.We call such a diagram a geometric representation for the matroid.Now we will define the projective geometry.Let V be a vector space over F. For each , ṽ w if v and w lie on the same 1-dimensional subspace of V.Then, ~ is an equivalence relation on V and { } 0 / Ṽ − is called the projective space of V or projective geometry and will be denoted by ( ) PG V .For a matroid M, delete all the loops from M and then, for each non-trivial parallel class X, delete all but one distinguished element of X, the matroid we obtain is called the simple matroid associated with M and is denoted by M  .Evidently the construction of ( ) V is analoguous to the construction of the simple matroid M  from a matroid M. It is clear that a matroid M is F-representable if and only if its associated simple matriod is F-representable.Hence, when we discuss representability questions, it is enough to concentrate on simple matroids. through v (Figure 1).Thus

Duals of Representable Matroids
Give a m n × matrix A by elementary row operations and interchanging two columns Hence, the dual of F-representable matroid is F-representable.For example, let [ ] 3 1 2 3 4 5 6 7 be a matrix over 2  .Then, we can see that

[ ]
M A is isomorphic to 7 F .

Minors
In this section, we define minors which are important to representability.For the definition of minor, we have to define contraction which is the dual of the operation of deletion.We can see that contraction for matroids generalizes the operation of contraction for graphs.Let M be a matroid on E and T be a subset of E. Then and we see that the contraction of a graphic matroid is the same as the matroid of the contracted graph, where we used Let T B be a basis for | there exists a basis of | such that .

T T M T I E T I B M I E T B M B I M
Proof.For the convenience, let's denote the equality by = =    .It is clear that and it was proved that ( ) ( ) 2) . 3) is obtained if we replace M by * M in the lefthand side of 1).□ Now, let A be a matroid over F and T be a subset of the set E of column levels of A.

We shall denote by \
A T the matrix obtained from A by deleting all the columns whose labels are in T. Clearly,

[ ] [ ] [ ]
Proof.It is enough to show that the second equation is true, because the first equation is clear.By using row and column swaps if necessary, A′ can be considered as the matrix in which the unique nonzero entry of e is in row 1 and column 1.Let I be a k-element subset of the ground set of [ ] M A′ such that e I ∉ .Then the set of columns labelled by I e ∪ is linearly independent if and only if the matrix B which has columns e I ∪ and the 1st column of it is the column corresponding to e has rank 1 k + .This is equivalent to the matrix deleted row 1 and column 1 of B has rank k and this is equivalent to the columns of / A e ′ labelled by I is linearly independent.Thus

Representability of P 8
Now, we shall describe the construction of representations for matroids.Two matrices , where r I is the r r × identity matrix.Given such a matrix, let its columns be labelled in order, 1  .Let B be the basis { } , , , r e e e  for M. Let X be the B -fundamental-circuit incidence matrix of M. And let columns of X be labelled by .Thus the task of finding an F-representation for M can be viewed as being one of finding the specific elements of F that correspond to the non-zero elements of # D .
We can see that most of the entries of D can be predetermined by the following Proposition 4.1.Before stating it, we shall require some preliminaries.
Let the rows of # D be indexed by 1 2 , , , r e e e  and its columns by By the above proposition, we can find the fields on which given matroid is representable.
Example 4.2.6 P is the matroid whose geometric representation is the following  We want to find the fields ( ) GF q in which the negative equations satisfy.It is easy to see that the equations have no solution if ( ) GF q is equal to 2  or 3  .In case of ( ) { } is the matrix over F with By taking a basis of ( )  P is the matroid of the matrix 1 2 3 4 5 6 7 P is representable over a field F if and only if , then the B-fundamental circuit incidence matrix of 7 P is By taking a basis of ( ) . Therefore, we proved that 7 P is representable over F if and only if Non F-representable matroid for which every proper minor is F-representable is called the excluded or forbidden minor for F-representability.Because a matroid M is F-representable if and only if all its minors are F-representable (Proposition 3.5), finding the complete set of excluded minors for F-representability is the solution for the F-representability.Since the duals of F-representable matroid are F-representable, the dual of an excluded minor for F-representability is an excluded minor for F-representability.
To find an excluded minor for 2  -representability, we need the following property: Proposition 4.5.Let F be a field and k be an integer exceeding 1.Then uniform matroid 2,k U is F-representable if and only if , where A is a 2 k × matrix.We can consider A as a matrix

□
From the above proposition, we can see that 2, 2 q U + and ( ) * 2, 2 , 2 q q q U U + + = are excluded minors for GF(q)-representability.In 1958, Tutte showed that set of excluded minors for GF(4)-representability. Oxley showed that the conjecture is wrong by showing 8

(C 1 ) 2 C are in  and 1 C and 2 C
∅ ∈ /  .(C 2 ) If 1C and are distinct members of  and r X of X to be the size of a maximal independent set of | X M .This function which is called the rank function of M satisfies the conditions (R 1 ), (R 2 ) and (R 3 ).This function is denoted by M r and

1 k
− and have coefficients in p  .There are exactly p choices for each of the k-coefficient of a member of S. Hence, k S p = .If we take p = 2 and k = 2 we get the field GF(4).Moreover, under addition and multiplication, both of which are performed modulo ( ) h ω , S forms a field namely

1 M and 2 M
ColA  is a matroid and this matroid is denoted by [ ] M A , where ColA is the set of all column vectors of A. Now, let G be a graph.Then ( ) M G is the matroid on the edge set ( ) E G with the set of all cycles of G as circuit.This matroid is called the cycle matroid of G. Two matroids are isomorphic, denoted by 1 2 the representation of X by points in m  , there are two identical points, or three collinear points, or four coplanar points, or five points in space.Hence the flats of M of ranks one, two, and three are represented geometrically by points, lines, and planar, respectively.We extend the use of diagram of affine matroid to represent arbitrary matroids of rank at most four.Generally, such diagrams are governed by the following rules.All loops are marked in a single inset.Parallel elements are represented by touching points.

)
PG V has dimension n and it is denoted by no nonzero elements except v on the 1-dimensional subspace of3   2
is called the contraction of T from M and also denoted by ( ) M E T ⋅ − .For easy understanding, let us see what it means in graphic matroids.Let G be a graph and same as G/3.Thus,

M
be the dual of a matroid M. Then the rank function * r of * M is given by that a contraction of F-representable matroid is F-representable.□Nowsuppose that e is the label of a non-zero column of A. Then, by pivoting on a non-zero entry of e, we can transform A into a matrix A′ in which the column labelled by e has single non-zero entry.In this case, / A e ′ will denote the matrix obtained from A′ by deleting the row and column containing the unique non-zero entry in e.Then, we have the following property.Proposition 3.5.


, the unique non-zero entry in column i of [ ] | r I D is in row i.Thus it is natural to label the rows of [ ] is called the B-fundamental circuit of k e .Let # D be the matrix obtained from D by replacing each non-zero entry of D by 1. Then the columns of # D are precisely the incidence vectors of the sets ( )


is an ordered k-tuple of non-zero elements of F, then M has an F-representation [ ] that, foreach i in { } 1, 2, , k , where the entry of 2 D corresponding to i b is i θ ([9][10]).

Figure 4
Figure 4. Let [ ] 3 | I D be a representation over
's check if the equations have a common solution.There are three cases− + − = − + − = ≠ .Therefore we showed that 6 P is representable over F if and only if

2 7 M
charF ≠ , then it is easy to see that [ ]

over 3 
. It's geometric representation is the following Figure 5. Lemma 4.4.7

2 k
are non-zero different elements of F. Thus, it should be and we can choose non-zero different ( ) − -elements of F.

8 P
be the matrix over 3  .Then, 8 P is the matroid [ ] is representable over a field F if and only if 2 From the first and fourth equation, we get b d = .Substituting b for d in the second and third equation, we get ( ) )-representability.In Examples 4.2 and 4.3, we can see that 6 P and 7 F − are not GF(4)-representable.It is not difficult to see that every proper minor of them is GF(4)-representable.Thus 6