The open polynomials of the finite topologies

zbMATH Open1346.11013arXiv1310.3605MaRDI QIDQ5404226

Publication date: 24 March 2014

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.

Full work available at URL: https://arxiv.org/abs/1310.3605

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zbMATH Keywords

limit theorem independence polynomial Fibonacci number unimodal sequence log-concave sequence centipede polynomial with real zeros

Mathematics Subject Classification ID

Other combinatorial number theory (11B75) Fibonacci and Lucas numbers and polynomials and generalizations (11B39)