A polynomial f is said to have the half-plane property if there is an open half-plane H, whose boundary contains the origin, such that f is non-zero whenever all the variables are in H. This paper answers several open questions regarding multivariate polynomials with the half-plane property and matroid theory.
* We prove that the support of a multivariate polynomial with the half-plane property is a jump system. This answers an open question posed by Choe, Oxley, Sokal and Wagner and generalizes their recent result claiming that the same is true whenever the polynomial is also homogeneous.
* We characterize multivariate multi-affine polynomial with real coefficients that have the half-plane property (with respect to the upper half-plane) in terms of inequalities. This is used to answer two open questions posed by Choe and Wagner regarding strongly Rayleigh matroids.
* We prove that the Fano matroid is not the support of a polynomial with the half-plane property. This is the first instance of a matroid which does not appear as the support of a polynomial with the half-plane property and answers a question posed by Choe et al.
We also discuss further directions and open problems.
A multivariate polynomial is stable if it is non-vanishing whenever all variables have positive imaginary parts. A matroid has the weak half-plane property (WHPP) if there exists a stable polynomial with support equal to the set of bases of the matroid. If the polynomial can be chosen with all of its nonzero coefficients equal to one then the matroid has the half-plane property (HPP). We describe a systematic method that allows us to reduce the WHPP to the HPP for large families of matroids. This method makes use of the Tutte group of a matroid. We prove that no projective geometry has the WHP…
We prove a new exchange property for bases of a matroid that generalizes the multiple symmetric exchange property. For every bases $B_1,\dots,B_k$ of a matroid and a subset $A_1\subset B_1$ there exist subsets $A_2\subset B_2,\dots,A_k\subset B_k$ such that all sets $(B_i\setminus A_i)\cup A_{i-1}$ achieved by a cyclic shift of $A_i$'s by one are bases.
The multiple exchange property for matroid bases is generalized for valuated matroids and M$^\natural$-concave set functions. The proof is based on the Fenchel-type duality theorem in discrete convex analysis. The present result has an implication in economics: The strong no complementarities (SNC) condition of Gul and Stacchetti is in fact equivalent to the gross substitutes (GS) condition of Kelso and Crawford.
We provide a weaker version of the generalized lifting property which holds in complete generality for all finite Coxeter groups, and we use it to show that every parabolic Bruhat interval of a finite Coxeter group is a Coxeter matroid. We also describe some combinatorial properties of the associated polytopes.
We investigate the strong Rayleigh property of matroids for which the basis enumerating polynomial is invariant under a Young subgroup of the symmetric group on the ground set. In general, the Grace-Walsh-Szegő theorem can be used to simplify the problem. When the Young subgroup has only two orbits, such a matroid is strongly Rayleigh if and only if an associated univariate polynomial has only real roots. When this polynomial is quadratic we get an explicit structural criterion for the strong Rayleigh property. Finally, if one of the orbits has rank two then the matroid is strongly Rayleigh if…
We prove the Lefschetz property for a certain class of finite-dimensional Gorenstein algebras associated to matroids. Our result implies the Sperner property of the vector space lattice. More generally, it is shown that the modular geometric lattice has the Sperner property. We also discuss the Gröbner fan of the defining ideal of our Gorenstein algebra.
Let $M$ be a matroid satisfying a matroidal analogue of the Cayley-Bacharach condition. Given a number $k \ge 2$, we show that there is no nontrivial bound on ranks of a $k$-tuple of flats covering the underlying set of $M$. This addresses a question of Levinson-Ullery motivated by earlier results which show that bounding the number of points satisfying the Cayley-Bacharach condition forces them to lie on low-dimensional linear subspaces. We also explore the general question what matroids satisfy the matroidal Cayley-Bacharach condition of a given degree and its relation to the geometry of gen…
Symmetries of geometric structures such as hyperplane arrangements, point configurations and polytopes have been studied extensively for a long time. However, symmetries of oriented matroids, a common combinatorial abstraction of them, are not understood well. In this paper, we aim to obtain a better understanding of symmetries of oriented matroids. First, we put focus on symmetries of matroids, and give a general construction that generates a $3$-dimensional point configuration with a matroidal symmetry that cannot be realized as a geometric symmetry. The construction is based on the observat…
We study the rank 4 linear matroid $M(H_4)$ associated with the 4-dimensional root system $H_4$. This root system coincides with the vertices of the 600-cell, a 4-dimensional regular solid. We determine the automorphism group of this matroid, showing half of the 14,400 automorphisms are geometric and half are not. We prove this group is transitive on the flats of the matroid, and also prove this group action is primitive. We use the incidence properties of the flats and the {\it orthoframes} of the matroid as a tool to understand these automorphisms, and interpret the flats geometrically.
Let $ E $ be a possibly infinite set and let $ M $ and $ N $ be matroids defined on $ E $. We say that the pair $ \{ M,N \} $ has the Intersection property if $ M $ and $ N $ share an independent set $ I $ admitting a bipartition $ I_M\sqcup I_N $ such that $ \mathsf{span}_M(I_M)\cup \mathsf{span}_N(I_N)=E $. The Matroid Intersection Conjecture of Nash-Williams says that every matroid pair has the Intersection property. The conjecture is known and easy to prove in the case when one of the matroids is uniform and it was shown by Bowler and Carmesin that the conjecture is implied by its special …
We investigate properties of Ehrhart polynomials for matroid polytopes, independence matroid polytopes, and polymatroids. In the first half of the paper we prove that for fixed rank their Ehrhart polynomials are computable in polynomial time. The proof relies on the geometry of these polytopes as well as a new refined analysis of the evaluation of Todd polynomials. In the second half we discuss two conjectures about the h^*-vector and the coefficients of Ehrhart polynomials of matroid polytopes; we provide theoretical and computational evidence for their validity.
In this paper we investigate the Ehrhart Theory of the independence matroid polytope of uniform matroids. It is proved that these polytopes have an Ehrhart polynomial with positive coefficients. To do that, we prove that indeed all half-open-hypersimplices are Ehrhart positive, and tile disjointly our polytope using them.
The Holt-Klee theorem says that the graph of a $d$-polytope, with edges oriented by a linear function on $P$ that is not constant on any edge, admits $d$ independent monotone paths from the source to the sink. We prove that the digraphs obtained from oriented matroid programs of rank $d+1$ on $n+2$ elements, which include those from $d$-polytopes with $n$ facets, admit $d$ independent monotone paths from source to sink if $d \le 4$. This was previously only known to hold for $d\le 3$ and $n\le 6$.
A matroid $\mathcal{M}$ on a set $E$ of elements has the $α$-partition property, for some $α>0$, if it is possible to (randomly) construct a partition matroid $\mathcal{P}$ on (a subset of) elements of $\mathcal{M}$ such that every independent set of $\mathcal{P}$ is independent in $\mathcal{M}$ and for any weight function $w:E\to\mathbb{R}_{\geq 0}$, the expected value of the optimum of the matroid secretary problem on $\mathcal{P}$ is at least an $α$-fraction of the optimum on $\mathcal{M}$. We show that the complete binary matroid, ${\cal B}_d$ on $\mathbb{F}_2^d$ does not satisfy the $α$-p…
We consider matroids with the property that every subset of the ground set of size $t$ is contained in both an $\ell$-element circuit and an $\ell$-element cocircuit; we say that such a matroid has the $(t,\ell)$-property. We show that for any positive integer $t$, there is a finite number of matroids with the $(t,\ell)$-property for $\ell<2t$; however, matroids with the $(t,2t)$-property form an infinite family. We say a matroid is a $t$-spike if there is a partition of the ground set into pairs such that the union of any $t$ pairs is a circuit and a cocircuit. Our main result is that if a su…
The Rayleigh property is a negative correlation property of electrical networks, which was generalized to matroids by Choe and Wagner. We prove that positroids, a class of matroid introduced by Postnikov which have seen many recent applications in both math and physics, have the Rayleigh property.
Rigidity and reconstruction in matroids of highly connected graphs
HUN-REN-ELTE Egerváry Research Group on Combinatorial Optimization and the MTA-ELTE Momentum Matroid Optimization Research Group（匈牙利研究与教育网络-厄特沃什·罗兰大学埃格瓦里组合优化研究组和匈牙利科学院-厄特沃什·罗兰大学动量拟阵优化研究组）；HUN-REN Alfréd Rényi Institute of Mathematics（匈牙利研究与教育网络阿尔弗雷德·雷尼数学研究所）
A polynomial P in n complex variables is said to have the "half-plane property" (or Hurwitz property) if it is nonvanishing whenever all the variables lie in the open right half-plane. Such polynomials arise in combinatorics, reliability theory, electrical circuit theory and statistical mechanics. A particularly important case is when the polynomial is homogeneous and multiaffine: then it is the (weighted) generating polynomial of an r-uniform set system. We prove that the support (set of nonzero coefficients) of a homogeneous multiaffine polynomial with the half-plane property is necessarily …
We prove that there is no polynomial $p(\cdot)$ with the property that a matroid $M$ can be determined to be either a lifted-graphic or frame matroid using at most $p(|M|)$ rank evaluations. This resolves two conjectures of Geelen, Gerards and Whittle (Quasi-graphic matroids, arXiv:1512.03005v1).
We introduce a new axiomatization of matroid theory that requires the elimination property only among modular pairs of circuits, and we present a cryptomorphic phrasing thereof in terms of Crapo's axioms for flats. This new point of view leads to a corresponding strengthening of the circuit axioms for oriented matroids.
This paper studies systems of polynomial equations that provide information about orientability of matroids. First, we study systems of linear equations over GF(2), originally alluded to by Bland and Jensen in their seminal paper on weak orientability. The Bland-Jensen linear equations for a matroid M have a solution if and only if M is weakly orientable. We use the Bland-Jensen system to determine weak orientability for all matroids on at most nine elements and all matroids between ten and twelve elements having rank three. Our experiments indicate that for small rank, about half the time, wh…
Generalizing the Multiple Exchange Property for Matroid Bases
Institute for Chemical Reaction Design and Discovery (ICReDD), Hokkaido University（北海道大学化学反应用设计发现研究所）；D3 Center, The University of Osaka（大阪大学D3中心）；Center for Advanced Intelligence Project, RIKEN（理化学研究所先进智能项目中心）；HUN-REN Alfréd Rényi Institute of Mathematics（匈牙利研究与教育网络阿尔弗雷德·雷尼数学研究所）；London School of Economics and Political Science（伦敦政治经济学院）
The multiple exchange property for matroid bases has recently been generalized for valuated matroids and M$^{\natural}$-concave set functions. This paper establishes a stronger form of this multiple exchange property that imposes a cardinality condition on the exchangeable subset. The stronger form immediately implies the defining exchange property of M$^{\natural}$-concave set functions, which was not the case with the recently established multiple exchange property without the cardinality condition.
We consider the relationship between a matroidal analogue of the degree $a$ Cayley-Bacharach property (finite sets of points failing to impose independent conditions on degree $a$ hypersurfaces) and geometric properties of matroids. If the matroid polytopes in question are nestohedra, we show that the minimal degree matroidal Cayley-Bacharach property denoted $MCB(a)$ is determined by the structure of the building sets used to construct them. This analysis also applies for other degrees $a$. Also, it does not seem to affect the combinatorial equivalence class of the matroid polytope. However, …
We show that in an ordered matroid the partial derivative \partial^{p+q}t/\partialx^p\partialyq of the Tutte polynomial is p!q! times the generating function of activities of subsets with corank p and nullity q. More generally, this property holds for the 3-variable Tutte polynomial of a matroid perspective.
We establish a convenient necessary and sufficient condition for a multiaffine real polynomial to be stable, and use it to verify that the half-plane property holds for seven small matroids that resisted the efforts of Choe, Oxley, Sokal, and Wagner [5].
In 2004, Choe, Oxley, Sokal and Wagner established a tight connection between matroids and multiaffine real stable polynomials. Recently, Branden used this theory and a polynomial coming from the Vamos matroid to disprove the generalized Lax conjecture. Here we present a 10-element extension of the Vamos matroid and prove that its basis generating polynomial is real stable (i.e. that the matroid has the half-plane property). We do this via large sums of squares computations and a criterion for real stability given by Wagner and Wei. Like the Vamos matroid, this matroid is not representable ove…
Anari, Gharan, and Vinzant proved (complete) log-concavity of the basis generating functions for all matroids. From the viewpoint of combinatorial Hodge theory, it is natural to ask whether these functions are "strictly" log-concave for simple matroids. In this paper, we show this strictness for simple graphic matroids, that is, we show that Kirchhoff polynomials of simple graphs are strictly log-concave. Our key observation is that the Kirchhoff polynomial {of a complete graph} can be seen as the (irreducible) relative invariant of a certain prehomogeneous vector space, which may be independe…
Let M be a matroid on ground set E. A subset l of E is called a `line' when its rank equals 1 or 2. Given a set L of lines, a `fractional matching' in (M,L) is a nonnegative vector x indexed by the lines in L, that satisfies a system of linear constraints, one for each flat of M. Fractional matchings were introduced by Vande Vate, who showed that the set of fractional matchings is a half-integer relaxation of the matroid matching polytope. It was shown by Chang et al. that a maximum size fractional matching can be found in polynomial time. In this paper we give a polynomial time algorithm to f…
Holt and Klee have recently shown that every (generic) LP orientation of the graph of a $d$-polytope satisfies a directed version of the $d$-connectivity property, i.e. there are $d$ internally disjoint directed paths from a unique source to a unique sink. We introduce two new classes HK and HK* of oriented matroids (OMs) by enforcing this property and its dual interpretation in terms of line shellings, respectively. Both classes contain all representable OMs by the Holt-Klee theorem. While we give a construction of an infinite family of non-HK* OMs, it is not clear whether there exists any no…
For all positive integers $t$ exceeding one, a matroid has the cyclic $(t-1,t)$-property if its ground set has a cyclic ordering $σ$ such that every set of $t-1$ consecutive elements in $σ$ is contained in a $t$-element circuit and $t$-element cocircuit. We show that if $M$ has the cyclic $(t-1,t)$-property and $|E(M)|$ is sufficiently large, then these $t$-element circuits and $t$-element cocircuits are arranged in a prescribed way in $σ$, which, for odd $t$, is analogous to how 3-element circuits and cocircuits appear in wheels and whirls, and, for even $t$, is analogous to how 4-element cir…
We characterise the following property by six obstructions: given a graphic matroid $M$ and a set $X$ of its elements, when is $M$ the cycle matroid of a graph $G$ such that $X$ is a connected edge set in $G$?
The Tutte polynomial of a graph or a matroid, named after W. T. Tutte, has the important universal property that essentially any multiplicative graph or network invariant with a deletion and contraction reduction must be an evaluation of it. The deletion and contraction operations are natural reductions for many network models arising from a wide range of problems at the heart of computer science, engineering, optimization, physics, and biology. Even though the invariant is #P-hard to compute in general, there are many occasions when we face the task of computing the Tutte polynomial for some …
We introduce the active partition of the ground set of an oriented matroid perspective (or quotient, or strong map) on a linearly ordered ground set. The reorientations obtained by arbitrarily reorienting parts of the active partition share the same active partition. This yields an equivalence relation for the set of reorientations of an oriented matroid perspective, whose classes are enumerated by coefficients of the Tutte polynomial, and a remarkable partition of the set of reorientations into boolean lattices, from which we get a short direct proof of a 4-variable expansion formula for the …
The Tutte polynomial is the most general invariant of matroids and graphs that can be computed recursively by deleting and contracting edges. We generalize this invariant to any class of combinatorial objects with deletion and contraction operations, associating to each such class a universal Tutte character by a functorial procedure. We show that these invariants satisfy a universal property and convolution formulae similar to the Tutte polynomial. With this machinery we recover classical invariants for delta-matroids, matroid perspectives, relative and colored matroids, generalized permutohe…
Partial-twuality polynomial interpolation for binary delta-matroids
School of Mathematical Sciences, Xiamen University（厦门大学数学科学学院）；School of Mathematics and Statistics, Lanzhou University（兰州大学数学与统计学院）；School of Mathematics and Statistics, Qinghai Minzu University（青海民族大学数学与统计学院）
Queen Mary University of London（伦敦大学玛丽女王学院）；San Francisco State University（旧金山州立大学）；Princeton University（普林斯顿大学）；University of Western Ontario（韦仕敦大学（西安大略大学））；Carnegie Mellon University（卡内基梅隆大学）；Korea Institute for Advanced Study（韩国高等研究院）；University of Wisconsin-Madison（威斯康星大学麦迪逊分校）
For a finite multigraph G, the reliability function of G is the probability R_G(q) that if each edge of G is deleted independantly with probability q then the remaining edges of G induce a connected spanning subgraph of G; this is a polynomial function of q. In 1992, Brown and Colbourn conjectured that for any connected multigraph G, if the complex number q is such that R_G(q)=0 then |q|<=1. We verify that this conjectured property of R_G(q) holds if G is a series-parallel network. The proof is by an application of the Hermite-Biehler Theorem and development of a theory of higher-order interla…
In this work, we study matroids over a domain and several classical combinatorial and algebraic invariants related. We define their Grothendieck-Tutte polynomial $T_{\mathcal{M}}(x,y)$, extending the definition given by Fink and Moci in 2016, and we show that such polynomial has the classical deletion-contraction property. Moreover, we study the set of independencies for a realizable matroid over a domain, generalizing the definition of \emph{poset of torsions} $Gr(\mathcal{M})$ given by the second author in 2017. This is a union of identical simplicial posets as for (quasi-)arithmetic matroid…
