42 research outputs found
One-adhesive polymatroids
Adhesive polymatroids were defined by F. Mat\'u\v{s} motivated by entropy
functions. Two polymatroids are adhesive if they can be glued together along
their joint part in a modular way; and are one-adhesive, if one of them has a
single point outside their intersection. It is shown that two polymatroids are
one-adhesive if and only if two closely related polymatroids have any
extension. Using this result, adhesive polymatroid pairs on a five-element set
are characterized
Amalgamation of real zero polynomials
With this article, we hope to launch the investigation of what we call the
real zero amalgamation problem. Whenever a polynomial arises from another
polynomial by substituting zero for some of its variables, we call the second
polynomial an extension of the first one. The real zero amalgamation problem
asks when two (multivariate real) polynomials have a common extension (called
amalgam) that is a real zero polynomial. We show that the obvious necessary
conditions are not sufficient. Our counterexample is derived in several steps
from a counterexample to amalgamation of matroids by Poljak and Turz\'ik. On
the positive side, we show that even a degree-preserving amalgamation is
possible in three very special cases with three completely different
techniques. Finally, we conjecture that amalgamation is always possible in the
case of two shared variables. The analogue in matroid theory is true by another
work of Poljak and Turz\'ik. This would imply a very weak form of the
Generalized Lax Conjecture.Comment: 24 page
On pseudomodular matroids and adjoints
AbstractThere are two concepts of duality in combinatorial geometry. A set theoretical one, generalizing the structure of two orthocomplementary vector spaces, and a lattice theoretical concept of an adjoint, that mimics duality between points and hyperplanes. The latter — usually called polarity — seems to make sense almost only in the linear case. In fact the only non-linear combinatorial geometries known to admit an adjoint were of rank 3. Moreover, N.E. Mnëv conjectured that in higher ranks there would exist no non-linear oriented matroid that has an oriented adjoint. At least with unoriented matroids this is not true. In this paper we present a class of rank-4 matroids with adjoint including a non-linear example