54 research outputs found
Field extensions, Derivations, and Matroids over Skew Hyperfields
We show that a field extension in positive characteristic
and elements for gives rise to a matroid on
ground set with coefficients in a certain skew hyperfield . This
skew hyperfield is defined in terms of and its Frobenius action
. The matroid underlying describes the
algebraic dependencies over among the , and itself
comprises, for each , the space of -derivations of
.
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-Pl\"ucker coordinates.Comment: Changed the signing convention for coordinates to better conform to
existing concepts in the literature (Tutte group, quasi-determinants
Algebraic matroids and Frobenius flocks
We show that each algebraic representation of a matroid in positive
characteristic determines a matroid valuation of , which we have named the
{\em Lindstr\"om valuation}. If this valuation is trivial, then a linear
representation of in characteristic can be derived from the algebraic
representation. Thus, so-called rigid matroids, which only admit trivial
valuations, are algebraic in positive characteristic if and only if they
are linear in characteristic .
To construct the Lindstr\"om valuation, we introduce new matroid
representations called flocks, and show that each algebraic representation of a
matroid induces flock representations.Comment: 21 pages, 1 figur
Asymptotics of Symmetry in Matroids
We prove that asymptotically almost all matroids have a trivial automorphism
group, or an automorphism group generated by a single transposition.
Additionally, we show that asymptotically almost all sparse paving matroids
have a trivial automorphism group.Comment: 10 page
On the number of matroids compared to the number of sparse paving matroids
It has been conjectured that sparse paving matroids will eventually
predominate in any asymptotic enumeration of matroids, i.e. that
, where denotes the number of
matroids on elements, and the number of sparse paving matroids. In
this paper, we show that We prove this by arguing that each matroid on 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 elements.
As a consequence of our result, we find that for some ,
asymptotically almost all matroids on elements have rank in the range .Comment: 12 pages, 2 figure
Perfect matroids over hyperfields
We investigate valuated matroids with an additional algebraic structure on
their residue matroids. We encode the structure in terms of representability
over stringent hyperfields.
A hyperfield is {\em stringent} if is a singleton unless
, for all . By a construction of Marc Krasner, each valued
field gives rise to a stringent hyperfield.
We show that if is a stringent skew hyperfield, then the vectors of any
weak matroid over are orthogonal to its covectors, and we deduce that weak
matroids over are strong matroids over . Also, we present vector axioms
for matroids over stringent skew hyperfields which generalize the vector axioms
for oriented matroids and valuated matroids.Comment: 19 page
Computing excluded minors for classes of matroids representable over partial fields
We describe an implementation of a computer search for the "small" excluded minors for a class of matroids representable over a partial field. Using these techniques, we enumerate the excluded minors on at most 15 elements for both the class of dyadic matroids, and the class of 2-regular matroids. We conjecture that there are no other excluded minors for the class of 2-regular matroids; whereas, on the other hand, we show that there is a 16-element excluded minor for the class of dyadic matroids.We describe an implementation of a computer search for the "small" excluded minors for a class of matroids representable over a partial field. Using these techniques, we enumerate the excluded minors on at most 15 elements for both the class of dyadic matroids, and the class of 2-regular matroids. We conjecture that there are no other excluded minors for the class of 2-regular matroids; whereas, on the other hand, we show that there is a 16-element excluded minor for the class of dyadic matroids
How to Design a Stable Serial Knockout Competition
We investigate a new tournament format that consists of a series of
individual knockout tournaments; we call this new format a Serial Knockout
Competition (SKC). This format has recently been adopted by the Professional
Darts Corporation. Depending on the seedings of the players used for each of
the knockout tournaments, players can meet in the various rounds (eg first
round, second round, ..., semi-final, final) of the knockout tournaments.
Following a fairness principle of treating all players equal, we identify an
attractive property of an SKC: each pair of players should potentially meet
equally often in each of the rounds of the SKC. If the seedings are such that
this property is indeed present, we call the resulting SKC stable. In this note
we formalize this notion, and we address the question: do there exist seedings
for each of the knockout tournaments such that the resulting SKC is stable? We
show, using a connection to the Fano plane, that the answer is yes for 8
players. We show how to generalize this to any number of players that is a
power of 2, and we provide stable schedules for competitions on 16 and 32
player
De Bruijn graphs and DNA graphs
In this paper we prove the NP-hardness of various recognition problems for subgraphs of De Bruijn graphs. In particular, the recognition of DNA graphs is shown to be NP-hard; DNA graphs are the vertex induced subgraphs of De Bruijn graphs over a four letter alphabet. As a consequence, two open questions from a recent paper by Blazewicz, Hertz, Kobler & de Werra [Discrete Applied Mathematics 98, 1999] are answered in the negative
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