On excluded minors for real-representability

AbstractWe show that for any infinite field K and any K-representable matroid N there is an excluded minor for K-representability that has N as a minor

Matroids with nine elements

We describe the computation of a catalogue containing all matroids with up to nine elements, and present some fundamental data arising from this cataogue. Our computation confirms and extends the results obtained in the 1960s by Blackburn, Crapo and Higgs. The matroids and associated data are stored in an online database, and we give three short examples of the use of this database.Comment: 22 page

Confinement of matroid representations to subsets of partial fields

Let M be a matroid representable over a (partial) field P and B a matrix representable over a sub-partial field P' of P. We say that B confines M to P' if, whenever a P-representation matrix A of M has a submatrix B, A is a scaled P'-matrix. We show that, under some conditions on the partial fields, on M, and on B, verifying whether B confines M to P' amounts to a finite check. A corollary of this result is Whittle's Stabilizer Theorem. A combination of the Confinement Theorem and the Lift Theorem from arXiv:0804.3263 leads to a short proof of Whittle's characterization of the matroids representable over GF(3) and other fields. We also use a combination of the Confinement Theorem and the Lift Theorem to prove a characterization, in terms of representability over partial fields, of the 3-connected matroids that have k inequivalent representations over GF(5), for k = 1, ..., 6. Additionally we give, for a fixed matroid M, an algebraic construction of a partial field P_M and a representation A over P_M such that every representation of M over a partial field P is equal to f(A) for some homomorphism f:P_M->P. Using the Confinement Theorem we prove an algebraic analog of the theory of free expansions by Geelen et al.Comment: 45 page

Towards a matroid-minor structure theory

This paper surveys recent work that is aimed at generalising the results and techniques of the Graph Minors Project of Robertson and Seymour to matroids

Representing some non-representable matroids

We extend the notion of representation of a matroid to algebraic structures that we call skew partial fields. Our definition of such representations extends Tutte's definition, using chain groups. We show how such representations behave under duality and minors, we extend Tutte's representability criterion to this new class, and we study the generator matrices of the chain groups. An example shows that the class of matroids representable over a skew partial field properly contains the class of matroids representable over a skew field. Next, we show that every multilinear representation of a matroid can be seen as a representation over a skew partial field. Finally we study a class of matroids called quaternionic unimodular. We prove a generalization of the Matrix Tree theorem for this class.Comment: 29 pages, 2 figure

Combinatorial and Computational Methods for the Properties of Homogeneous Polynomials

In this manuscript, we provide foundations of properties of homogeneous polynomials such as the half-plane property, determinantal representability, being weakly determinantal, and having a spectrahedral hyperbolicity cone. One of the motivations for studying those properties comes from the ``generalized Lax conjecture'' stating that every hyperbolicity cone is spectrahedral. The conjecture has particular importance in convex optimization and has curious connections to other areas. We take a combinatorial approach, contemplating the properties on matroids with a particular focus on operations that preserve these properties. We show that the spectrahedral representability of hyperbolicity cones and being weakly determinantal are minor-closed properties. In addition, they are preserved under passing to the faces of the Newton polytopes of homogeneous polynomials. We present a proved-to-be computationally feasible algorithm to test the half-plane property of matroids and another one for testing being weakly determinantal. Using the computer algebra system Macaulay2 and Julia, we implement these algorithms and conduct tests. We classify matroids on at most 8 elements with respect to the half-plane property and provide our test results on matroids with 9 elements. We provide 14 matroids on 8 elements of rank 4, including the Vámos matroid, that are potential candidates for the search of a counterexample for the conjecture.:1 Background 1 1.1 Some Properties of Homogeneous Polynomials . . . . . . . . . . 1 Hyperbolic Polynomials . . . . . . . . . . . . . . . . . . . . . . 1 The Half-Plane Property and Stability . . . . . . . . . . . . . . 8 Determinantal Representability . . . . . . . . . . . . . . . . . . 15 Spectrahedral Representability . . . . . . . . . . . . . . . . . . 19 1.2 Matroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . 25 Some Operations on Matroids . . . . . . . . . . . . . . . . . . . 29 The Half-Plane Property of Matroids . . . . . . . . . . . . . . . 36 2 Some Operations 43 2.1 Determinantal Representability of Matroids . . . . . . . . . . . 43 A Criterion for Determinantal Representability . . . . . . . . . 46 2.2 Spectrahedral Representability of Matroids . . . . . . . . . . . 50 2.3 Matroid Polytopes . . . . . . . . . . . . . . . . . . . . . . . . . 54 Newton Polytopes of Stable Polynomials . . . . . . . . . . . . . 59 3 Testing the Properties: an Algorithm 61 The Half-Plane Property . . . . . . . . . . . . . . . . . . . . . . 61 Being SOS-Rayleigh and Weak Determinantal Representability 65 4 Test Results on Matroids on 8 and 9 Elements 71 4.1 Matroids on 8 Elements . . . . . . . . . . . . . . . . . . . . . . 71 SOS-Rayleigh and Weakly Determinantal Matroids . . . . . . . 76 4.2 Matroids on 9 Elements . . . . . . . . . . . . . . . . . . . . . . 80 5 Conclusion and Future Perspectives 85 5.1 Spectrahedral Matroids . . . . . . . . . . . . . . . . . . . . . . 86 5.2 Non-negative Non-SOS Polynomials . . . . . . . . . . . . . . . 88 5.3 Completing the Classification of Matroids on 9 Elements and More 89 Bibliography 9

Kinser inequalities and related matroids

Kinser developed a hierarchy of inequalities dealing with the dimensions of certain spaces constructed from a given quantity of subspaces. These inequalities can be applied to the rank function of a matroid, a geometric object concerned with dependencies of subsets of a ground set. A matroid which is representable by a matrix with entries from some finite field must satisfy each of the Kinser inequalities. We provide results on the matroids which satisfy each inequality and the structure of the hierarchy of such matroids.Comment: 82 pages, 7 figures. MSc thesi