The open polynomials of the finite topologies

2013/10/14 by Moussa Benoumhani, Benoumhani, Moussa
Mathematics · #11B39 #11B75 #Combinatorics (math.CO) #FOS: Mathematics #General Topology (math.GN) #Number Theory (math.NT) #math.CO #math.GN #math.NT #msc:11B39 #msc:11B75

paper · pdf · doi:10.48550/arxiv.1310.3605

14 pages

arxiv created 2013/10/14 · arxiv updated 2013/10/15

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.

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