On the relation between hyperrings and fuzzy rings

We construct a full embedding of the category of hyperfields into Dress's category of fuzzy rings and explicitly characterize the essential image --- it fails to be essentially surjective in a very minor way. This embedding provides an identification of Baker's theory of matroids over hyperfields with Dress's theory of matroids over fuzzy rings (provided one restricts to those fuzzy rings in the essential image). The embedding functor extends from hyperfields to hyperrings, and we study this extension in detail. We also analyze the relation between hyperfields and Baker's partial demifields

Field extensions, derivations, and matroids over skew hyperfields

We show that a field extension $K\subseteq L$ in positive characteristic $p$ and elements $x_e\in L$ for $e\in E$ gives rise to a matroid $M^\sigma$ on ground set $E$ with coefficients in a certain skew hyperfield $L^\sigma$. This skew hyperfield $L^\sigma$ is defined in terms of $L$ and its Frobenius action $\sigma:x\mapsto x^p$. The matroid underlying $M^\sigma$ describes the algebraic dependencies over $K$ among the $x_e\in L$ , and $M^\sigma$ itself comprises, for each $m\in \mathbb{Z}^E$, the space of $K$-derivations of $K\left(x_e^{p^{m_e}}: e\in E\right)$. The theory of matroid representation over hyperfields was developed by Baker and Bowler for commutative hyperfields. We partially extend their theory to skew hyperfields. To prove the duality theorems we need, we use a new axiom scheme in terms of quasi-Plucker coordinates

On the evaluation at (−ι,ι)(-\iota,\iota) of the Tutte polynomial of a binary matroid

Vertigan has shown that if $M$ is a binary matroid, then $|T_M(-\iota,\iota)|$, the modulus of the Tutte polynomial of $M$ as evaluated in $(-\iota, \iota)$, can be expressed in terms of the bicycle dimension of $M$. In this paper, we exactly determine $T_M(-\iota,\iota)$, and show how to evaluate this number in polynomial time. In particular, we describe how the argument of the complex number $T_M(-\iota,\iota)$ depends on a certain Z mod four valued quadratic form that is canonically associated with M

On the number of matroids

We consider the problem of determining $m_n$, the number of possible matroids on $n$ elements. The best known lower bound on $m_n$ is due to Knuth (1974) who showed that $\log \log m_n$ is at least $n-3/2\log n-1$. On the other hand, Piff (1973) showed that $\log\log m_n\leq n-\log n+\log\log n +O(1)$, and it has been conjectured since that the right answer is perhaps closer to Knuth's bound. Here, we show that this is indeed the case. In particular, we show that $\log\log m_n\leq n-3/2\log n+2\log\log n+O(1)$, which matches Knuth's lower bound up to second order terms. To this end, we show some new structural properties of non-bases in a matroid and use them to give a compressed representation of matroids

Reconstructing a phylogenetic level-1 network from quartets

We describe a method that will reconstruct an unrooted binary phylogenetic level-1 network on n taxa from the set of all quartets containing a certain fixed taxon, in O(n^3) time. We also present a more general method which can handle more diverse quartet data, but which takes O(n^6) time. Both methods proceed by solving a certain system of linear equations over the two-element field GF(2) . For a general dense quartet set, i.e. a set containing at least one quartet on every four taxa, our O(n^6) algorithm constructs a phylogenetic level-1 network consistent with the quartet set if such a network exists and returns an O(n^2) -sized certificate of inconsistency otherwise. This answers a question raised by Gambette, Berry and Paul regarding the complexity of reconstructing a level-1 network from a dense quartet set, and more particularly regarding the complexity of constructing a cyclic ordering of taxa consistent with a dense quartet set

Perfect matroids over hyperfields

A hyperfield $H$ is stringent if $a\boxplus b$ is a singleton unless $a=-b$, for all $a,b\in H$. By a construction of Marc Krasner, each valued field gives rise to a stringent hyperfield. We show that if $H$ is a stringent skew hyperfield, then weak matroids over $H$ are strong matroids over $H$. Also, we present vector axioms for matroids over stringent skew hyperfields which generalize the vector axioms for oriented matroids and valuated matroids

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'= 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 [Whi99]. A combination of the Confinement Theorem and the Lift Theorem from [PZ] leads to a short proof of Whittle's characterization of the matroids representable over GF(3) and other fields [Whi97]. 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 PM and a representation A over PM such that every representation of M over a partial field P is equal to ¿(A) for some homomorphism ¿:PM¿P. Using the Confinement Theorem we prove an algebraic analog of the theory of free expansions by Geelen et al

Enumerating matroids of fixed rank

It has been conjectured that asymptotically almost all matroids are sparse paving, i.e. that $s(n) \sim m(n)$, where $m(n)$ denotes the number of matroids on a fixed groundset of size $n$, and $s(n)$ the number of sparse paving matroids. In an earlier paper, we showed that $\log s(n) \sim \log m(n)$. The bounds that we used for that result were dominated by matroids of rank $r\approx n/2$. In this paper we consider the relation between the number of sparse paving matroids $s(n,r)$ and the number of matroids $m(n,r)$ on a fixed groundset of size $n$ of fixed rank $r$. In particular, we show that $\log s(n,r) \sim \log m(n,r)$ whenever $r\ge 3$, by giving asymptotically matching upper and lower bounds. Our upper bound on $m(n,r)$ relies heavily on the theory of matroid erections as developed by Crapo and Knuth, which we use to encode any matroid as a stack of paving matroids. Our best result is obtained by relating to this stack of paving matroids an antichain that completely determines the matroid. We also obtain that the collection of essential flats and their ranks gives a concise description of matroids. It has been conjectured that asymptotically almost all matroids are sparse paving, i.e. that s(n)∼m(n) , where m(n) denotes the number of matroids on a fixed groundset of size n , and s(n) the number of sparse paving matroids. In an earlier paper, we showed that logs(n)∼logm(n) . The bounds that we used for that result were dominated by matroids of rank r≈n/2 . In this paper we consider the relation between the number of sparse paving matroids s(n,r) and the number of matroids m(n,r) on a fixed groundset of size n of fixed rank r . In particular, we show that logs(n,r)∼logm(n,r) whenever r≥3 , by giving asymptotically matching upper and lower bounds. \u3cbr/\u3eOur upper bound on m(n,r) relies heavily on the theory of matroid erections as developed by Crapo and Knuth, which we use to encode any matroid as a stack of paving matroids. Our best result is obtained by relating to this stack of paving matroids an antichain that completely determines the matroid. \u3cbr/\u3eWe also obtain that the collection of essential flats and their ranks gives a concise description of matroids

On the number of matroids compared to the number of sparse paving matroids

t has been conjectured that sparse paving matroids will eventually predominate in any asymptotic enumeration of matroids, i.e. that limn→∞sn/mn=1limn→∞sn/mn=1, where mnmn denotes the number of matroids on nn elements, and snsn the number of sparse paving matroids. In this paper, we show that limn→∞logsnlogmn=1. limn→∞log⁡snlog⁡mn=1. We prove this by arguing that each matroid on nn elements has a faithful description consisting of a stable set of a Johnson graph together with a (by comparison) vanishing amount of other information, and using that stable sets in these Johnson graphs correspond one-to-one to sparse paving matroids on nn elements. As a consequence of our result, we find that for all β>ln22−−−−√=0.5887⋯β>ln⁡22=0.5887⋯, asymptotically almost all matroids on nn elements have rank in the range n/2±βn−−√n/2±βn