4,888 research outputs found
Hyperfield extensions, characteristic one and the Connes-Consani plane connection
Inspired by a recent paper of Alain Connes and Catherina Consani which
connects the geometric theory surrounding the elusive field with one element to
sharply transitive group actions on finite and infinite projective spaces
("Singer actions"), we consider several fudamental problems and conjectures
about Singer actions. Among other results, we show that virtually all infinite
abelian groups and all (possibly infinitely generated) free groups act as
Singer groups on certain projective planes, as a corollary of a general
criterion. We investigate for which fields the plane
(and more generally the
space ) admits a Singer
group, and show, e.g., that for any prime and any positive integer ,
cannot admit Singer groups. One of the
main results in characteristic , also as a corollary of a criterion which
applies to many other fields, is that with a positive even integer, cannot admit Singer groups.Comment: 25 pages; submitted (June 2014). arXiv admin note: text overlap with
arXiv:1406.544
Secret-Sharing Matroids need not be Algebraic
We combine some known results and techniques with new ones to show that there
exists a non-algebraic, multi-linear matroid. This answers an open question by
Matus (Discrete Mathematics 1999), and an open question by Pendavingh and van
Zwam (Advances in Applied Mathematics 2013). The proof is constructive and the
matroid is explicitly given
A convex combinatorial property of compact sets in the plane and its roots in lattice theory
K. Adaricheva and M. Bolat have recently proved that if and are
circles in a triangle with vertices , then there exist and such that is included in the convex hull
of . One could say disks instead of
circles. Here we prove the existence of such a and for the more general
case where and are compact sets in the plane such that is
obtained from by a positive homothety or by a translation. Also, we give
a short survey to show how lattice theoretical antecedents, including a series
of papers on planar semimodular lattices by G. Gratzer and E. Knapp, lead to
our result.Comment: 28 pages, 7 figure
On the number of simple arrangements of five double pseudolines
We describe an incremental algorithm to enumerate the isomorphism classes of
double pseudoline arrangements. The correction of our algorithm is based on the
connectedness under mutations of the spaces of one-extensions of double
pseudoline arrangements, proved in this paper. Counting results derived from an
implementation of our algorithm are also reported.Comment: 24 pages, 16 figures, 6 table
Combinatorial geometries of field extensions
We classify the algebraic combinatorial geometries of arbitrary field
extensions of transcendence degree greater than 4 and describe their groups of
automorphisms. Our results and proofs extend similar results and proofs by
Evans and Hrushovski in the case of algebraically closed fields. The
classification of projective planes in algebraic combinatorial geometries in
arbitrary fields of characteristic zero will also be given.Comment: 11 pages, 1 figur
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