h-Vectors of matroids and logarithmic concavity

Let M be a matroid on E, representable over a field of characteristic zero. We show that h-vectors of the following simplicial complexes are log-concave:

1.

The matroid complex of independent subsets of E.     

Patroids

A matroid M over a set E of elements is semiseparated by a partition {S¹, S²} of E iff rank E = rank S¹ + rank S² + 1. Such a semiseparation defines in each Sⁱ a pair of matroids or patroid Pⁱ = (Mⁱ, mⁱ); the two patroids P¹, P² weld to form M. The operations of removing and contracting a non-degenerate element of a matroid produce a patroid. The properties of patroids, their bases, and circuits are discussed.     

Maximum nullity of outerplanar graphs and the path cover number

Let G=(V,E) be a graph with V={1,2,…,n}. Define S(G) as the set of all n×n real-valued symmetric matrices A=[a_ij] with a_ij≠0,i≠j if and only if ij∈E. By M(G) we denote the largest possible nullity of any matrix A∈S(G). The path cover number of a graph G, denoted P(G), is the minimum number of vertex disjoint paths occurring as induced subgraphs of G which cover all the vertices of G.There has been some success with relating the path cover number of a graph to its maximum nullity. Johnson and Duarte [5], have shown that for a tree T,M(T)=P(T). Barioli et al. [2], show that for a unicyclic graph G,M(G)=P(G) or M(G)=P(G)-1. Notice that both families of graphs are outerplanar. We show that for any outerplanar graph G,M(G)?P(G). Further we show that for any partial 2-path G,M(G)=P(G).     

On the unique representability of spikes over prime fields

For an integer n?3, a rank-n matroid is called an n-spike if it consists of n three-point lines through a common point such that, for all k in {1,2,…,n-1}, the union of every set of k of these lines has rank k+1. Spikes are very special and important in matroid theory. Wu [On the number of spikes over finite fields, Discrete Math. 265 (2003) 261-296] found the exact numbers of n-spikes over fields with 2, 3, 4, 5, 7 elements, and the asymptotic values for larger finite fields. In this paper, we prove that, for each prime number p, a GF(p) representable n-spike is only representable on fields with characteristic p provided that n?2p-1. Moreover, M is uniquely representable over GF(p).     

Spanning trails containing given edges

A graph G is Eulerian-connected if for any u and v in V(G), G has a spanning (u,v)-trail. A graph G is edge-Eulerian-connected if for any e^′ and e^″ in E(G), G has a spanning (e^′,e^″)-trail. For an integer r?0, a graph is called r-Eulerian-connected if for any X⊆E(G) with |X|?r, and for any , G has a spanning (u,v)-trail T such that X⊆E(T). The r-edge-Eulerian-connectivity of a graph can be defined similarly. Let θ(r) be the minimum value of k such that every k-edge-connected graph is r-Eulerian-connected. Catlin proved that θ(0)=4. We shall show that θ(r)=4 for 0?r?2, and θ(r)=r+1 for r?3. Results on r-edge-Eulerian connectivity are also discussed.     

Galois groups of multivariate Tutte polynomials

The multivariate Tutte polynomial $\hat{Z}_{M}$ of a matroid M is a generalization of the standard two-variable version, obtained by assigning a separate variable v _e to each element e of the ground set E. It encodes the full structure of M. Let v={v _e } _e??E , let K be an arbitrary field, and suppose M is connected. We show that $\hat{Z}_{M}$ is irreducible over K(v), and give three self-contained proofs that the Galois group of $\hat{Z}_{M}$ over K(v) is the symmetric group of degree n, where n is the rank of M. An immediate consequence of this result is that the Galois group of the multivariate Tutte polynomial of any matroid is a direct product of symmetric groups. Finally, we conjecture a similar result for the standard Tutte polynomial of a connected matroid.     

Equivariant Reduction to Torus of a Principal Bundle

Let M be an irreducible projective variety, over an algebraically closed field k of characteristic zero, equipped with an action of a connected algebraic group S over k. Let E _G be a principal G-bundle over M equipped with a lift of the action of S on M, where G is a connected reductive linear algebraic group. Assume that E _G admits a reduction of structure group to a maximal torus TG. We give a necessary and sufficient condition for the existence of a T-reduction of E _G which is left invariant by the action of S on E _G .     

Connected matroids with a small circumference

Lemos and Oxley proved that if M is a connected matroid with |E(M)|⩾3r(M), then M has a circuit C such that M⧹C is connected. In this paper, we shall improve this result proving that for a simple and connected matroid M, if r(M)⩾7 and |E(M)|⩾3r(M)−3, then M has a circuit C such that M⧹C is connected. To prove this result, we shall construct all the connected matroids having circumference at most five, with the exception of those which are 3-connected and have rank five.     

Representations of matroids and free resolutions for multigraded modules

Let k be a field, let R=k[x₁,…,xm] be a polynomial ring with the standard Zm-grading (multigrading), let L be a Noetherian multigraded R-module, and let be a finite free multigraded presentation of L over R. Given a choice S of a multihomogeneous basis of E, we construct an explicit canonical finite free multigraded resolution T_•(Φ,S) of the R-module L. In the case of monomial ideals our construction recovers the Taylor resolution. A main ingredient of our work is a new linear algebra construction of independent interest, which produces from a representation ? over k of a matroid M a canonical finite complex of finite dimensional k-vector spaces T_•(?) that is a resolution of Ker?. We also show that the length of T_•(?) and the dimensions of its components are combinatorial invariants of the matroid M, and are independent of the representation map ?.     

Non-regular simplicial matroids

Let (ks) denote the set of all k-element-subsets of a finite set S. A k-simplical matroid on a subset E of (ks) is a binary matroid the circuit of which are simplicial complexes {X1,…Xm} ? E with boundary 0 (mod 2). The k-simplical matroid on (ks) is called the full simplicial matroid Gk(S). The polygon matroid on the edges of a finite graph is 2-simplicial. Polygon-matroids and their duals are regular. The dual of Gk(S) is Gn?k(S) if the cardinnlity of S is n. More details on simplicial matroids can be found in [3, Chapter 6] and also in [4, pp. 180–181].Welsh asked if every simplicial matroid is regular. We prove that this is not the case, for all full k-simplicial matroids Gk(S) with 3?k?n?3 are non-regular (n is the cardinality of S). This result has also been proved σy R. Cordovil and M. Las Vergnas recently. Their proof is different from our proof, which is somewhat shorter.     

Extremal critically connected matroids

A connected matroid M is called a critically connected matroid if the deletion of any one element from M results in a disconnected matroid. We show that a critically connected matroid of rank n, n≥3, can have at most 2n?2 elements. We also show that a critically connected matroid of rank n on 2n?2 elements is isomorphic to the forest matroid of K₂, n?2.     

Module derivation problem for inverse semigroups

The derivation problem for a locally compact group G asserts that each bounded derivation from L ¹(G) to L ¹(G) is implemented by an element of M(G). Recently a simple proof of this result was announced. We show that basically the same argument with some extra manipulations with idempotents solves the module derivation problem for inverse semigroups, asserting that for an inverse semigroup S with set of idempotents E and maximal group homomorphic image G _S , if E acts on S trivially from the left and by multiplication from the right, any bounded module derivation from ? ¹(S) to ? ¹(G _S ) is inner.     

Root numbers and ranks in positive characteristic

For a global field K and an elliptic curve E_η over K(T), Silverman's specialization theorem implies rank(E_η(K(T)))?rank(E_t(K)) for all but finitely many t∈P¹(K). If this inequality is strict for all but finitely many t, the elliptic curve E_η is said to have elevated rank. All known examples of elevated rank for K=Q rest on the parity conjecture for elliptic curves over Q, and the examples are all isotrivial.Some additional standard conjectures over Q imply that there does not exist a non-isotrivial elliptic curve over Q(T) with elevated rank. In positive characteristic, an analogue of one of these additional conjectures is false. Inspired by this, for the rational function field K=κ(u) over any finite field κ with characteristic ≠2, we construct an explicit 2-parameter family E_c,d of non-isotrivial elliptic curves over K(T) (depending on arbitrary c,d∈κ^×) such that, under the parity conjecture, each E_c,d has elevated rank.     

Bi-cycle extendable through a given set in balanced bipartite graphs

Let G=(X,Y;E) be a balanced bipartite graph of order 2n. The path-cover numberpc(H) of a graph H is the minimum number of vertex-disjoint paths that use up all the vertices of H. S⊆V(G) is called a balanced set of G if |S∩X|=|S∩Y|. In this paper, we will give some sufficient conditions for a balanced bipartite graph G satisfying that for every balanced set S, there is a bi-cycle of every length from |S|+2pc(〈S〉) up to 2n through S.     

On some properties of base-matroids

Let M=(E,F) be a rank-n matroid on a set E and B one of its bases. A closed set θ⊆E is saturated with respect to B, or B-saturated, when |θ∩B|=r(θ), where r(θ) is the rank of θ.The collection of subsets I of E such that |I∩θ|?r(θ), for every closed B-saturated set θ, turns out to be the family of independent sets of a new matroid on E, called base-matroid and denoted by M_B. In this paper we prove some properties of M_B, in particular that it satisfies the base-axiom of a matroid.Moreover, we determine a characterization of the matroids M which are isomorphic to M_B for every base B of M.Finally, we prove that the poset of the closed B-saturated sets ordered by inclusion is isomorphic to the Boolean lattice B_n.     

The basis monomial ring of a matroid

We define the basis monomial ring M_G of a matroid G and prove that it is Cohen-Macaulay for finite G. We then compute the Krull dimension of M_G, which is the rank over Q of the basis-point incidence matrix of G, and prove that dim B_G ≥ dim M_G under a certain hypothesis on coordinatizability of G, where B_G is the bracket ring of G.     

Matroids with given restrictions and contractions

If M is a matroid on a set S and if X is a subset of S, then there are two matroids on X induced by M: namely, the restriction and the contraction of M onto X. Necessary and sufficient conditions are obtained for two matroids on the same set to be of this form and an analogous result is obtained when (X₁,…, X_t) is a partition of S. The corresponding results when all the matroids are binary are also obtained.     

The connectivity and minimum degree of circuit graphs of matroids

Let G be the circuit graph of any connected matroid M with minimum degree 5（G）. It is proved that its connectivity κ（G） ≥2｜E（M） - B（M）｜ - 2. Therefore 5（G） ≥ 2｜E（M） - B（M）｜ - 2 and this bound is the best possible in some sense.     

Functorial topologies and finite-index subgroups of abelian groups

In the general context of functorial topologies, we prove that in the lattice of all group topologies on an abelian group, the infimum between the Bohr topology and the natural topology is the profinite topology. The profinite topology and its connection to other functorial topologies is the main objective of the paper. We are particularly interested in the poset C(G) of all finite-index subgroups of an abelian group G, since it is a local base for the profinite topology of G. We describe various features of the poset C(G) (its cardinality, its cofinality, etc.) and we characterize the abelian groups G for which C(G)?{G} is cofinal in the poset of all subgroups of G ordered by inclusion. Finally, for pairs of functorial topologies T, S we define the equalizer E(T,S), which permits to describe relevant classes of abelian groups in terms of functorial topologies.     

A chain theorem for internally 4-connected binary matroids

Let M be a matroid. When M is 3-connected, Tutte's Wheels-and-Whirls Theorem proves that M has a 3-connected proper minor N with |E(M)−E(N)|=1 unless M is a wheel or a whirl. This paper establishes a corresponding result for internally 4-connected binary matroids. In particular, we prove that if M is such a matroid, then M has an internally 4-connected proper minor N with |E(M)−E(N)|?3 unless M or its dual is the cycle matroid of a planar or Möbius quartic ladder, or a 16-element variant of such a planar ladder.