The open polynomials of the finite topologies
Combinatorics
2013-10-15 v1 General Topology
Number Theory
Abstract
Let T be a topology on the finite set Xn. We consider the open polynomial associated with the topology T. Its coefficients are the cardinality of open sets of size j=0,...,n. J. Brown [4] asked when this polynomial has only real zeros. We prove that this polynomial has real zeros, only in the trivial case where T is the discrete topology. Then, we weaken Brown's question: for which topology this polynomial is log-concave, or at least unimodal? A partial answer is given. Precisely, we prove that if the topology has a large number of open sets, then its open polynomial is unimodal.
Cite
@article{arxiv.1310.3605,
title = {The open polynomials of the finite topologies},
author = {Moussa Benoumhani},
journal= {arXiv preprint arXiv:1310.3605},
year = {2013}
}
Comments
14 pages