Relaxations of GF$(4)$-representable matroids

We consider the GF$(4)$-representable matroids with a circuit-hyperplane such that the matroid obtained by relaxing the circuit-hyperplane is also GF$(4)$-representable. We characterize the structure of these matroids as an application of structure theorems for the classes of $U_{2,4}$-fragile and $\{U_{2,5},U_{3,5}\}$-fragile matroids. In addition, we characterize the forbidden submatrices in GF$(4)$-representations of these matroids.


Introduction
Lucas [9] determined the binary matroids that have a circuit-hyperplane whose relaxation yields another binary matroid. Truemper [16], and independently, Oxley and Whittle [13], did the same for ternary matroids. In this paper, we solve the corresponding problem for quaternary matroids. We give both a structural characterization and a characterization in terms of forbidden submatrices.
Truemper [16] used the structure of circuit-hyperplane relaxations of binary and ternary matroids to give new proofs of the excluded-minor characterizations for the classes of binary, ternary, and regular matroids. It is natural to ask if Truemper's techniques can be extended to give excluded-minor characterizations for classes of quaternary matroids. The main results of this paper can be viewed as a first step towards answering this question.
Our structural characterization can be summarized as follows. A matroid has path width 3 if there is an ordering (e 1 , e 2 , . . . , e n ) of its ground set such that {e 1 , e 2 , . . . , e t } is a 3-separating set for all t ∈ {1, 2, . . . , n}. Theorem 1. Let M and M be GF(4)-representable matroids such that M is obtained from M by relaxing a circuit-hyperplane. Then M has path width 3.
In fact, our main result, Theorem 35, describes precisely how the matroids in Theorem 1 of path width 3 can be constructed using the generalized ∆-Y exchange of [12] and the notion of gluing a wheel onto a triangle from [2]. Our description uses the structure of U 2,4 -fragile matroids from [10] and the structure of {U 2,5 , U 3,5 }-fragile matroids from [3].
In future work, we hope to obtain a description of these matroids that is independent of the notion of fragility. Specifically, we would like to characterize the representations of these matroids. As a step in this direction, we describe minimal GF(4)-representations of matroids with a circuit-hyperplane whose relaxation is not GF(4)-representable. Note that the proof uses the excluded-minor characterization of the class of GF(4)-representable matroids. The setup for this result is as follows.
Let M be a GF(4)-representable matroid on E with a circuit-hyperplane X. Choose e ∈ X and f ∈ E − X such that (X − e) ∪ f is a basis of M . Then M = M [I|C] for a quaternary matrix C of the following block form.
In the above matrix, A is an (X − e) × ((E − X) − f ) matrix, and we have scaled so that every non-zero entry in the row labelled by f and the column labelled by e is 1. Let M be the matroid obtained from M by relaxing the circuit-hyperplane X. We call the matrix C a reduced representation of M . If M is GF(4)-representable, then we can find a reduced representation C of M in the following block form. We have scaled the rows and columns of the matrix such that the entry C f e = ω ∈ GF(4) − {0, 1}, and the remaining entries in row f and column e are all 1. The following theorem is our characterization in terms of forbidden submatrices. Theorem 2. Let M and C be constructed as described above. There is a reduced representation C of the above form for M if and only if, up to permuting rows and columns, A and A T have no submatrix in the following list, where x, y, z denote distinct non-zero elements of GF(4). x y x y y 0 This paper is organized as follows. In the next section, we collect some results on connectivity and circuit-hyperplane relaxation. In Section 3, we prove a fragility theorem. In Section 4, we describe the structure of the {U 2,5 , U 3,5 }-fragile matroids. In Section 5, we prove the structural characterization. In Section 6, we reduce the proof of Theorem 2 to a finite computer check. This check, carried out using SageMath, can be found in the Appendix [4].

Circuit-hyperplane relaxations and connectivity
We assume the reader is familiar with the fundamentals of matroid theory. Any undefined matroid terminology will follow Oxley [11]. Let M be a matroid on E, and let B(M ) denote the collection of bases of M . If M has a circuit-hyperplane X, then B(M ) = B(M ) ∪ {X} is the collection of bases of a matroid M on E. We say that M is obtained from M by relaxing the circuit-hyperplane X. We list here a number of useful results on circuithyperplane relaxation.
The following elementary results are originally from [8]. Dually, when f ∈ X, (ii) M\f = M \f and, unless M has f as a loop, M /f is obtained from M/f by relaxing the circuit-hyperplane X − f of the latter.
For a set N of matroids, we say that a matroid M has an N -minor if M has an N -minor for some N ∈ N . We say M is N -fragile if M has an N -minor and, for each element e of M , at most one matroid in {M\e, M/e} has an N -minor. We say an element e of an N -fragile matroid M is nondeletable if M\e has no N -minor; the element e is noncontractible if M/e has no N -minor.
The following lemma is an immediate consequence of Lemma 4.
the electronic journal of combinatorics 25(2) (2018), #P2.53 Lemma 5. Let X be a circuit-hyperplane of a matroid M , and let M be the matroid obtained from M by relaxing X. If N is a set of matroids such that M has an Nminor but M has no N -minor, then M is N -fragile. Moreover, X is a basis of M whose elements are nondeletable such that the elements of the cobasis E(M ) − X are noncontractible.
We use the following connectivity result.
Lemma 6. [11,Proposition 8.4.2] Let M be a matroid that is obtained by relaxing a circuit-hyperplane of a matroid M . If M is n-connected, then M is n-connected.
Kahn [8] proved the following result on the representability of a circuit-hyperplane relaxation.
Lemma 7. Let M be a matroid that is obtained by relaxing a circuit-hyperplane of a matroid M . If M is connected, then M is non-binary.
We use the following definition of the rank function of the 2-sum from [7]. Let M 1 and M 2 be matroids with at least two elements such that The next three results on 2-sums and minors of 2-sums are well known. We can now describe the structure of circuit-hyperplanes in matroids of low connectivity. We omit the straightforward proof of the next lemma.
Lemma 11. Let M be a GF(4)-representable matroid with a circuit-hyperplane H. If M is not connected, then M ∼ = U 1,m ⊕ U n−1,n for some positive integers m and n.
We now work towards a description of the 2-separations of a connected matroid in which the relaxation of some circuit-hyperplane is GF(4)-representable.
the electronic journal of combinatorics 25(2) (2018), #P2.53 Lemma 12. Let M be a matroid with a circuit-hyperplane X. If A is a non-trivial parallel class of M , then either A ⊆ E − X, or A = X and |A| = 2.
Proof. If A ∩ X and A ∩ (E − X) are both non-empty, then there is a circuit {x, y} contained in A such that x ∈ X and y ∈ E − X. But E − X is a cocircuit of M , so this is a contradiction to orthogonality. Thus either A ∩ X or A ∩ (E − X) is empty. In the case that A ∩ (E − X) is empty, there is a circuit {x, y} contained in A that is also contained in the circuit X, so X = A = {x, y}.
For the next result, we say that M is 3-connected up to series and parallel classes if M is connected and, for any 2-separation (X, Y ) of M , either X or Y is a series class or a parallel class.
Lemma 13. Let M be a GF(4)-representable matroid with a circuit-hyperplane X such that the matroid M obtained from M is also GF(4)-representable. If M is connected but not 3-connected, then M is 3-connected up to series and parallel classes.
Proof. Assume that M has a 2-separation (S, T ) where neither side is a series or parallel class. Then M has a 2-sum decomposition of the form M = N ⊕ 2 N for some N and N with E(N ) ∩ E(N ) = {p}, where neither N nor N is a circuit or cocircuit.
First suppose that the circuit X of M has the form (C ∪ C ) − p, where C is a circuit of N , and C is a circuit of N while p ∈ C ∩ C . Then and r M (X) = r N (C) + r N (C ) − 1. We may therefore assume that C is a spanning circuit of N , and hence that E(N ) = C because the hyperplane X is closed. Therefore N is a circuit, a contradiction. By symmetry, it remains to consider the case when X is a circuit of N \p. Then r(X) r(N ). Since X is a hyperplane of M , and r(M ) = r(N )+r(N )−1, it follows that r(N ) 2. Since N is not a cocircuit, we deduce that r(N ) = 2. Then r(M ) = r(N ) + 1, so r(X) = r(N ) = r(N \p). Since N is not a circuit we deduce that si(N ) ∼ = U 2,m for some m 4. Moreover, p is not in a non-trivial parallel class in N otherwise X is not a hyperplane of M .
Switching to M * , we see that r M * (N ) = |X| + r(N ) − r(M ) = r(N ) = 2. As above, it follows that co(N ) ∼ = U n−1,n+1 for some n 3. Moreover, p is not in a non-trivial series class in N . Let X 1 consist of one representative of each series class of N , and let the electronic journal of combinatorics 25(2) (2018), #P2.53 Y 1 consist of one representative of each parallel class of N . By contracting elements of X − X 1 and deleting elements of (E(M ) − X) − Y 1 , we obtain U n−1,n+1 ⊕ 2 U 2,m as a minor of M for some n 3 and m 4. Moreover, by Lemma 4, X 1 is a circuit-hyperplane of this minor whose relaxation is GF(4)-representable. Thus X 1 ⊆ E(U n−1,n+1 ). Contract n − 3 elements from X 1 and delete m − 4 elements from Y 1 to get U 2,4 ⊕ 2 U 2,4 . Relaxing a circuit-hyperplane of this minor gives P 6 which is not GF(4)-representable (see [11,Proposition 6.5.8]), a contradiction.

A fragility theorem
We will use the following consequence of Geelen, Oxley, Vertigan, and Whittle [6,Theorem 8.4].
Theorem 14. Let M and M be GF(4)-representable matroids with the properties that M is connected, M is 3-connected, and M is obtained from M by relaxing a circuithyperplane.
We can now prove the main result of this section.
Theorem 15. Let M and M be GF(4)-representable matroids such that M is connected, M is 3-connected, and M is obtained from M by relaxing a circuit-hyperplane X. Then M is either U 2,4 -fragile or {U 2,5 , U 3,5 }-fragile. Moreover, X is a basis of M whose elements are nondeletable such that the elements of the cobasis E(M ) − X are noncontractible.
Proof. First assume that M has no {U 2,5 , U 3,5 }-minor. By Lemma 7 and Theorem 14 (i), M has a U 2,4 -minor and M has no U 2,4 -minor. Then it follows from Lemma 5 that M is U 2,4 -fragile, and M has a basis X whose elements are nondeletable such that the elements of the cobasis E(M ) − X are noncontractible.
We may now assume that M has a {U 2,5 , U 3,5 }-minor. Suppose that M also has a {U 2,5 , U 3,5 }-minor, and assume that M is a minor-minimal matroid with respect to the hypotheses; that is, we assume that M has no proper minor M 0 such that M 0 is connected, M 0 has a {U 2,5 , U 3,5 }-minor, and M 0 has a circuit-hyperplane whose relaxation M 0 is 3connected, GF(4)-representable, and has a {U 2,5 , U 3,5 }-minor.
Proof of 16. Suppose that M has an element e ∈ E(M ) − X such that M\e has a {U 2,5 , U 3,5 }-minor. If M\e is 3-connected, then we have a contradiction to the minimality of M . Therefore, by Lemma 13, M\e is 3-connected up to series and parallel pairs. Suppose that A is a non-trivial parallel class of M\e. Suppose A ⊆ X. Then A = X and |A| = 2 by Lemma 12, so we deduce that M\e is a parallel extension of U 2,5 and hence the electronic journal of combinatorics 25(2) (2018), #P2.53 that M \e has a U 2,6 -minor, a contradiction to the fact that the matroid M obtained from M by relaxing X is GF(4)-representable. Thus A ⊆ E(M\e) − X by Lemma 12. By duality, any non-trivial series class of M\e must be contained in X. Then, by Lemma 10, the matroid M 0 obtained from M\e by deleting all but one element of every non-trivial parallel class and contracting all but one element of every non-trivial series class has a {U 2,5 , U 3,5 }-minor. We deduce from Lemma 13 that M 0 is 3-connected. Then M 0 contradicts the minimality of M . Therefore M\e has no {U 2,5 , U 3,5 }-minor for all e ∈ E(M )−X, and, by duality, M/e has no {U 2,5 , U 3,5 }-minor for all e ∈ X, so M is {U 2,5 , U 3,5 }-fragile. This completes the proof of 16.
Since M has a {U 2,5 , U 3,5 }-minor, it follows from Theorem 14 (ii) that M has a U 3,6minor, that is, M /C\D ∼ = U 3,6 for some subsets C and D. If C ⊆ X and D ⊆ E(M )−X, then it follows from Lemma 4 that U 3,6 can be obtained from M/C\D by relaxing the circuit-hyperplane X − C. Hence M/C\D ∼ = P 6 , a contradiction because M/C\D is GF(4)-representable but P 6 is not. Therefore C ∩ (E(M ) − X) or D ∩ X is nonempty, so M/C\D = M /C\D ∼ = U 3,6 by Lemma 4. This is a contradiction to 16 because any minor of M must also be {U 2,5 , U 3,5 }-fragile, but for any e, both U 3,6 \e and U 3,6 /e have a {U 2,5 , U 3,5 }-minor. We conclude that M has no {U 2,5 , U 3,5 }-minor. It now follows from Lemma 5 that M is {U 2,5 , U 3,5 }-fragile, and that M has a basis X whose elements are nondeletable such that the elements of the cobasis E(M ) − X are noncontractible. 4 The structure of {U 2,5 , U 3,5 }-fragile matroids 4

.1 Partial Fields and Constructions
We briefly state the necessary material on partial fields. For a more thorough treatment, we refer the reader to [14].
A partial field is a pair P = (R, G), where R is a commutative ring with unity, and G is a subgroup of the units of R with −1 ∈ G. A matrix with entries in G is a Pmatrix if det(D) ∈ G ∪ {0} for any square submatrix D of A. We use X to denote the multiplicative subgroup of R generated by the subset X.
A rank-r matroid M on the ground set E is P-representable if there is an r × |E| P-matrix A such that, for each r × r submatrix D, the determinant of D is nonzero if and only if the corresponding subset of E is a basis of M . When this occurs, we write The 2-regular partial field is defined as follows.
Lemma 17. The class of U 2 -representable matroids is contained in the class of GF(4)- To describe the structure of {U 2,5 , U 3,5 }-fragile matroids as in [3], we need two constructions: the generalized ∆-Y exchange, and gluing on wheels. For a more thorough treatment of these constructions, we refer the reader to [12] and [2].
Loosely speaking, the operations of generalized ∆-Y exchange and gluing on wheels both involve gluing matroids together along a common restriction. Let M 1 and M 2 be matroids with a common restriction A, where A is a modular flat of M 1 . The generalized parallel connection of M 1 and M 2 along A, denoted P A (M 1 , M 2 ), is the matroid obtained by gluing M 1 and M 2 along A. It has ground set E(M 1 ) ∪ E(M 2 ), and a set F is a flat of To define the generalized ∆-Y exchange formally, we first recall the following definition of a family of matroids Θ k from [12]. For k 3, fix a basis B = {b 1 , b 2 , . . . , b k } of the rank-k projective geometry P G(k − 1, R), and choose a line L of P G(k − 1, R) that is freely placed relative to B. If follows from modularity that, for each i, the hyperplane spanned by B − {b i } meets L; we let a i be the point of intersection. Let A = {a 1 , a 2 , . . . , a k }, and let Θ k be the matroid obtained by restricting P G(k − 1, R) to the set A ∪ B. Note that the matroid Θ k has A as a modular k-point segment A, so the generalized parallel connection of Θ k and M along A is well-defined. If the k-element segment A is coindependent in M , then we define the matroid ∆ A (M ) to be the matroid obtained from P A (Θ k , M )\A by relabeling the elements of E(Θ k ) − A by A in the natural way, and we say that ∆ A (M ) is obtained from M by performing a generalized ∆-Y exchange on A. For a matroid M with an independent cosegment A, a generalized Y -∆ exchange on A, denoted by ∇ A (M ), is defined to be the matroid (∆ A (M * )) * .
We use the following results on representability and the minor operations. Let M be a {U 2,5 , U 3,5 }-fragile matroid. A segment S of M is allowable if S is coindependent and some element of S is nondeletable. A cosegment C of M is allowable if the segment C of M * is allowable. In [3], it was shown that we can obtain a new {U 2,5 , U 3,5 }fragile U 2 -representable matroid from an old {U 2,5 , U 3,5 }-fragile U 2 -representable matroid by performing a generalized ∆-Y exchange on an allowable segment. We will prove an analogous result for {U 2,5 , U 3,5 }-fragile GF(4)-representable matroids with no {P − 8 , F = 7 , (F = 7 ) * }-minor. Let U be the class of GF(4)-representable matroids with no {U 2,5 , U 3,5 }-minor. The class of sixth-root-of-unity matroids is the class of matroids that are representable over both GF(3) and GF(4). Semple and Whittle [15,Theorem 5.2] showed that U is the class of matroids that can be obtained by taking direct sums and 2-sums of binary and sixth-root-of-unity matroids.
Lemma 24. Let M be a matroid in the class U. If M is obtained from M by performing a generalized ∆-Y exchange or a generalized Y -∆ exchange, then M ∈ U.
Proof. Suppose that there exists a matroid M ∈ U with a coindependent segment A such that ∆ A (M ) ∈ U. Among all counterexamples, suppose that M has been chosen so that |E(M )| is as small as possible. Suppose M is 3-connected. Since any 3-connected member of U is either a binary or sixth-root-of-unity matroid, this also holds for ∆ A (M ) by Lemma 18. Hence ∆ A (M ) ∈ U, contradicting the assumption that M is a counterexample. Therefore  -wheel onto (a, b, c). We also say that M * is obtained from N * by gluing a wheel onto the triad T . Suppose that T 1 , T 2 , . . . , T n are ordered triples whose underlying sets are triangles of M . We say M can be obtained from M by gluing wheels the electronic journal of combinatorics 25(2) (2018), #P2.53 onto T 1 , T 2 , . . . , T n if, for some subset J of {1, 2, . . . , n}, M can be obtained from M by a sequence of moves, where each move consists of gluing an r j -wheel onto T j for j ∈ J. Note that the spoke elements of a triangle in this sequence may only be deleted as part of the gluing operation when they do not appear in any subsequent triangle in the sequence. by gluing an r-wheel onto (a, b, c)

Path sequences
We can now describe a family of {U 2,5 , U 3,5 }-fragile GF(4)-representable matroids with no {P − 8 , F = 7 , (F = 7 ) * }-minor obtained by performing generalized ∆-Y exchanges and gluing on wheels. In fact, the matroids in this family are U 2 -representable and were first described in [3]. Each matroid in this family has a {X 8 , Y 8 , Y * 8 }-minor, and an associated path of 3-separations that we need to describe in order to define the family.
We call the set X ⊆ E(M ) fully closed if X is closed in both M * and M . The full closure of X, denoted fcl M (X), is the intersection of all fully closed sets containing X. The full closure of X can be obtained from X by repeatedly taking closure and coclosure until no new elements are added. We call X a path-generating set if X is a 3-separating set of M such that fcl M (X) = E(M ). A path-generating set X thus gives rise to a natural path of 3-separating sets (P 1 , . . . , P m ), where P 1 = X and each step P i is either the closure or coclosure of the 3-separating set P 1 ∪ · · · ∪ P i−1 .
Let X be an allowable cosegment of the {U 2,5 , U 3,5 }-fragile matroid M . A matroid Q is an allowable series extension of M along X if M = Q/Z and, for every element z of Z, there is some element x of X such that x is {U 2,5 , U 3,5 }-contractible in M and z is in series with x in Q. We also say that Q * is an allowable parallel extension of M * along X.
Let N be a matroid with a path-generating allowable segment or cosegment A. We say that M is obtained from N by a ∆-∇-step along A if, up to duality, M is obtained from N by performing a non-empty allowable parallel extension along A, followed by a generalized ∆-Y exchange on A.
Let X 8 be the matroid obtained from U 2,5 by choosing a 4-element segment C, adding a point in parallel with each of three distinct points of C, then performing a generalized ∆-Y -exchange on C (see Figure 1). In what follows, S will be the elements of the 4-element segment of X 8 , and C the elements of the 4-element cosegment of X 8 , so E(X 8 ) = S ∪ C. We will build matroids from X 8 by performing a sequence of ∆-∇-steps along A ∈ {S, C}. Note that, in such matroids, each of S and C can be either a segment or a cosegment. Note in (P2) that each ∆-∇-step described in (a) increases the number of elements by at least one, and that the wheels in (b) are only glued onto allowable subsets of 4-element segments or cosegments.
We say that a path sequence M 1 , . . . , M n describes a matroid M if M n ∼ = M . We also say that M is a matroid described by a path sequence if there is some path sequence that describes M . Let P denote the class of matroids such that M ∈ P if and only if there is some path sequence M 1 , . . . , M n that describes a matroid M such that M can be obtained from M by some, possibly empty, sequence of allowable parallel and series extensions. Since X 8 is self-dual, it is easy to see that the sequence of dual matroids M * 1 , . . . , M * n of a path sequence M 1 , . . . , M n is also a path sequence. Thus the class P is closed under duality.
We denote by Y 8 the unique matroid obtained from X 8 by performing a Y -∆-exchange on an allowable triad (see Figure 1). We will prove the following result.
The proof of Theorem 27 closely follows the proof of [3,Corollary 4.3]. The strategy is to show that a minor-minimal counterexample has at most 12 elements. Let M be a GF(4)-representable Suppose that M is a minimum-sized matroid that is not in the class P. Then M is 3-connected because P is closed under series and parallel extensions. Moreover, the dual M * is also not in P because P is closed under duality. Thus, by the Splitter Theorem and duality, we may assume there is some element x of M such that M\x the electronic journal of combinatorics 25(2) (2018), #P2.53 is also a 3-connected GF(4)-representable {U 2,5 , U 3,5 }-fragile matroid with an {X 8 , Y 8 , Y * 8 }minor but no {P − 8 , F = 7 , (F = 7 ) * }-minor. By the assumption that M is minimum-sized with respect to being outside the class P, it follows that M\x ∈ P. Thus M\x is described by a path sequence M 1 , . . . , M n . The next lemma [3, Lemma 6.3] identifies the three possibilities for the position of x in M relative to the path of 3-separations associated with M 1 , . . . , M n .
Lemma 28. Let M and M\x be 3-connected {U 2,5 , U 3,5 }-fragile matroids. If M\x is described by a path sequence with associated path of 3-separations P, then either: (i) there is some 3-separation (X, Y ) displayed by P such that x ∈ cl(X) and x ∈ cl(Y ); or (ii) there is some 3-separation (X, Y ) displayed by P such that x / ∈ cl(X) and x / ∈ cl(Y ); or (iii) for each 3-separation (R, G) of M displayed by P, there is some X ∈ {R, G} such that x ∈ cl M (X) and x ∈ cl * M (X). The

Fan extensions
The following theorem describes the structure of the matroids with no {X 8 , Y 8 , Y * 8 }-minor. Note that M 9,9 is the rank-4 matroid on 9 elements in Figure 2. The matroid M 7,1 is the 7-element matroid that is obtained from Y 8 by deleting the unique point that is contained in the two 4-element segments of Y 8 . We label the points of a triangle of M 7,1 by {1, 2, 3} as in Figure 2. We can now show that if M and M are GF(4)-representable matroids such that M is obtained from M by relaxing a circuit-hyperplane, then M has path width 3.
Proof of Theorem 1. If M is not connected, then it follows from Lemma 11 that M has path width 3. We may therefore assume that M is connected. Then, by Lemma 13, M can be obtained from a matroid in Theorem 35 (a) -(f) by performing some, possibly empty, sequence of series or parallel extensions. The result now follows from the fact that all the matroids in Theorem 35 (a) -(f) have path width 3.

Forbidden submatrices
In this section, we will prove our second characterization, Theorem 2. Let M be a GF(4)representable matroid with a circuit-hyperplane X. Choose e ∈ X and f ∈ E − X such that B = (X − e) ∪ f is a basis of M . Then we can find a reduced GF(4)-representation of M in block form, Here A is an (X − e) × ((E − X) − f ) matrix over GF(4), and we have scaled so that every non-zero entry in the row labelled by f and the column labelled by e is 1. We denote by A ij the entry in row i and column j of A.
Let M be the matroid obtained from M by relaxing the circuit-hyperplane X. If M is GF(4)-representable, then we can find a reduced representation of M in block form, We have scaled the rows and columns of the matrix such that the entry in the row labelled by f and column labelled by e is ω ∈ GF(4) − {0, 1}, and every remaining entry in row e and column f is a 1.
We omit the straightforward proof of the following lemma.
Lemma 36. A ij = 0 if and only if A ij = 0.
Next we show that the only non-zero entries of A are 1 and ω.

Proof. Suppose
The following lemma on diagonal submatrices will be used frequently. We can now identify all of the forbidden submatrices. We use Lemma 38 to identify the first such matrix in the following lemma. where a, b, c are distinct non-zero entries, which is a contradiction to Lemma 37.
We now use Lemma 38 and Lemma 39 to find several more forbidden submatrices.
Lemma 41. A has no submatrices of the following forms, where x, y, and z are distinct non-zero entries.
Proof. Suppose A has the submatrix (i). By applying Lemma 38 to the rows and the first column, we deduce that the corresponding submatrix of A has the form a a 0 a 0 a , where a is a non-zero entry, a contradiction of Lemma 39. Suppose A has the submatrix (ii). By applying Lemma 38 to the rows and the first column, and since A has at most two distinct non-zero entries by Lemma 37, we deduce that the corresponding submatrix of A has the form a a 0 a 0 b , where a and b are the two non-zero entries of A , a contradiction to Lemma 39. The proofs for (iii) and (iv) are similar to that for (ii). We omit the details. Suppose A has the submatrix (v). Then, by two applications of Lemma 39, the corresponding submatrix of A must have the form for some non-zero entry a. This is a contradiction to Lemma 38.
the electronic journal of combinatorics 25(2) (2018), #P2.53 Suppose A has the submatrix (vi). By Lemma 39, the corresponding submatrix of A must be a diagonal matrix with distinct non-zero entries, a contradiction to Lemma 37.
Suppose A has the submatrix (vii). Applying Lemma 39 to the two submatrices the form x 0 0 y , it follows that the corresponding submatrix of A is for some a, which is a contradiction to Lemma 39. Suppose A has the submatrix (viii). Then the corresponding submatrix of A is Lemma 42. A has no submatrices of the following forms, where x, y, and z are distinct non-zero entries: Proof. Suppose A has the submatrix (i). Then, adjoining e and f , we see that C has the following submatrix with non-zero determinant. where {a, b} = {1, ω} by Lemma 37. This gives a contradiction because this submatrix of C has zero determinant. A similar proof handles (ii). Suppose A has the submatrix (iii). Then, by Lemma 38, in the corresponding submatrix of A , the entries in the first row are the same and the entries in the second row are different. But, by Lemma 37, there are only two distinct non-zero entries in A , so the entries are the same in one of the columns of A , which is a contradiction to Lemma 38.
Suppose A has the submatrix (iv). Note that this submatrix has zero determinant. By Lemma 38, the corresponding submatrix of A must have the following form. Finally, we find two more 3 × 3 forbidden submatrices of A.
Lemma 43. A has no submatrices of the following forms, where x, y, and z are distinct non-zero entries: Proof. Suppose that A has the submatrix (i). Then, adjoining e and f , we see that C has the submatrix     x y x 1 y y 0 1 for distinct a, b ∈ {1, ω}. This submatrix of C has non-zero determinant, a contradiction. Suppose that A has the submatrix (ii). Note that the determinant of this submatrix is not zero. By Lemma 37 and Lemma 38, the corresponding submatrix of A is for distinct a, b ∈ {1, ω}. This submatrix of A has zero determinant, which is a contradiction.
To prove the main theorem of this section, we need the following theorem [5, Theorem 5.1].
g in both D and the circuit-hyperplane X of M , then M\g = M \g by Lemma 4, so M also has an N -minor, contradicting the fact that M is GF(4)-representable. We deduce that D ⊆ E(M ) − X, and dually, C ⊆ X. Now if |D| 2, then there is some element g in both D and E(M ) − (X ∪ f ), so relaxing the circuit-hyperplane X of M\g gives M \g that is not GF(4)-representable, which contradicts the minimality of M . Therefore |D| 1, and by a dual argument, there is no element g in both C and X − e, so |C| 1. Since we know, by Theorem 44, that |E(N )| 8, it now follows that |E(M )| 10. The computations in the Appendix [4] show that M must have a submatrix from the above list.