Codes, arrangements, matroids, and their polynomial links

Codes, arrangements, matroids, and their polynomial links Many mathematical objects are closely related to each other. While studying certain aspects of a mathematical object, one tries to find a way to "view" the object in a way that is most suitable for a specific problem. Or, in other words, one tries to find the best way to model the problem. Many related fields of mathematics have evolved from one another this way. In practice, it is very useful to be able to transform a problem into other terminology: it gives a lot more available knowledge and that can be helpful to solve a problem. This thesis deals with various closely related fields in discrete mathematics, starting from linear error-correcting codes and their weight enumerator. We can generalize the weight enumerator in two ways, to the extended and generalized weight enumerators. The set of generalized weight enumerators is equivalent to the extended weight enumerator. Summarizing and extending known theory, we define the two-variable zeta polynomial of a code and its generalized zeta polynomial. These polynomials are equivalent to the extended and generalized weight enumerator of a code. We can determine the extended and generalized weight enumerator using projective systems. This calculation is explicitly done for codes coming from finite projective and affine spaces: these are the simplex code and the first order Reed-Muller code. As a result we do not only get the weight enumerator of these codes, but it also gives us information on their geometric structure. This is useful information in determining the dimension of geometric designs. To every linear code we can associate a matroid that is representable over a finite field. A famous and well-studied polynomial associated to matroids is the Tutte polynomial, or rank generating function. It is equivalent to the extended weight enumerator. This leads to a short proof of the MacWilliams relations for the extended weight enumerator. For every matroid, its flats form a geometric lattice. On the other hand, every geometric lattice induces a simple matroid. The Tutte polynomial of a matroid determines the coboundary polynomial of the associated geometric lattice. In the case of simple matroids, this becomes a two-way equivalence. Another polynomial associated to a geometric lattice (or, more general, to a poset) is the Möbius polynomial. It is not determined by the coboundary polynomial, neither the other way around. However, we can give conditions under which the Möbius polynomial of a simple matroid together with the Möbius polynomial of its dual matroid defines the coboundary polynomial. The proof of these relations involves the two-variable zeta polynomial, that can be generalized from codes to matroids. Both matroids and geometric lattices can be truncated to get an object of lower rank. The truncated matroid of a representable matroid is again representable. Truncation formulas exist for the coboundary and Möbius polynomial of a geometric lattice and the spectrum polynomial of a matroid, generalizing the known truncation formula of the Tutte polynomial of a matroid. Several examples and counterexamples are given for all the theory. To conclude, we give an overview of all polynomial relations

Intrinsic Volumes of Polyhedral Cones: A combinatorial perspective

The theory of intrinsic volumes of convex cones has recently found striking applications in areas such as convex optimization and compressive sensing. This article provides a self-contained account of the combinatorial theory of intrinsic volumes for polyhedral cones. Direct derivations of the General Steiner formula, the conic analogues of the Brianchon-Gram-Euler and the Gauss-Bonnet relations, and the Principal Kinematic Formula are given. In addition, a connection between the characteristic polynomial of a hyperplane arrangement and the intrinsic volumes of the regions of the arrangement, due to Klivans and Swartz, is generalized and some applications are presented.Comment: Survey, 23 page

Mixing for Smooth Time-Changes of General Nilflows

We consider irrational nilflows on any nilmanifold of step at least $2$. We show that there exists a dense set of smooth time-changes such that any time-change in this class which is not measurably trivial gives rise to a mixing nilflow. This in particular reproves and generalizes to any nilflow (of step at least $2$) the main result proved in [AFU] for the special class of Heisenberg (step $2$) nilflows, and later generalized in [Rav2] to a class of nilflows of arbitrary step which are isomorphic to suspensions of higher-dimensional linear toral skew-shifts.Comment: 36 pages, 1 figur

Combinatorial Techniques in the Galois Theory of p-Extensions

A major open problem in current Galois theory is to characterize those profinite groups which appear as absolute Galois groups of various fields. Obtaining detailed knowledge of the structure of quotients and subgroup filtrations of Galois groups of p-extensions is an important step toward a solution. We illustrate several techniques for counting Galois p-extensions of various fields, including pythagorean fields and local fields. An expression for the number of extensions of a formally real pythagorean field having Galois group the dihedral group of order 8 is developed. We derive a formula for computing the Fp-dimension of an n-th graded piece of the Zassenhaus filtration for various finitely generated pro-p groups, including free pro-p groups, Demushkin groups and their free pro-p products. Several examples are provided to illustrate the importance of these dimensions in characterizing pro-p Galois groups. We also show that knowledge of small quotients of pro-p Galois groups can provide information regarding the form of relations among the group generators

A Tutte decomposition for matrices and bimatroids

AbstractWe develop a Tutte decomposition theory for matrices and their combinatorial abstractions, bimatroids. As in the graph or matroid case, this theory is based on a deletion–contraction decomposition. The contribution from the deletion, derived by an inclusion–exclusion argument, consists of three terms. With one more term contributed from the contraction, the decomposition has four terms in general. There are universal decomposition invariants, one of them being a corank–nullity polynomial. Under a simple change of variables, the corank–nullity polynomial equals a weighted characteristic polynomial. This gives an analog of an identity of Tutte. Applications to counting and critical problems on matrices and graphs are given

Distinguishing graphs by their left and right homomorphism profiles

We introduce a new property of graphs called ‘q-state Potts unique-ness’ and relate it to chromatic and Tutte uniqueness, and also to ‘chromatic–flow uniqueness’, recently studied by Duan, Wu and Yu. We establish for which edge-weighted graphs H homomor-phism functions from multigraphs G to H are specializations of the Tutte polynomial of G, in particular answering a question of Freed-man, Lovász and Schrijver. We also determine for which edge-weighted graphs H homomorphism functions from multigraphs G to H are specializations of the ‘edge elimination polynomial’ of Averbouch, Godlin and Makowsky and the ‘induced subgraph poly-nomial’ of Tittmann, Averbouch and Makowsky. Unifying the study of these and related problems is the notion of the left and right homomorphism profiles of a graph.Ministerio de Educación y Ciencia MTM2008-05866-C03-01Junta de Andalucía FQM- 0164Junta de Andalucía P06-FQM-0164

Uniformization, accessory parameters and modular forms

Several topics related to modular forms and to the accessory parameter problem for the uniformization of hyperbolic Riemann surfaces are discussed. In the first part of the thesis we present an algorithm for the computation of the accessory parameters for the Fuchsian uniformization of certain punctured spheres. Then, via modular forms of rational weight, we show that the knowledge of the uniformizing differential equation leads to the complete knowledge of the ring of modular forms $M_*(Gamma)$ and of its Rankin-Cohen structure. In the second part of the thesis, a new operator $partial_ ho$ is defined on the space of quasimodular forms $widetilde{M}_*(Gamma)$ from an infinitesimal deformation of the uniformizing differential equation. It is shown that $partial_ ho$ can be described in terms of well-known derivations on $widetilde{M}_*(Gamma)$ and certain integrals of weight four-cusp forms; the relation between the operator $partial_ ho$ and a classical construction in Teichm"uller theory is discussed. The functions $partial_ ho{g},,ginwidetilde{M}_*(Gamma),$ motivate the study and the introduction of a new class of functions, called emph{extended modular forms}. Extended modular forms are defined as certain components of vector-valued modular forms with respect to symmetric tensor representations. Apart from the functions $partial_ ho{g},$ examples of extended modular forms are: Eichler integrals, more general iterated integrals of modular forms, and elliptic multiple zeta values