On Polynomials Associated with Finite Topologies - Moussa Benoumhani; Brahim Chaourar (Axioms)
Let <semantics>τ</semantics> be a topology on the finite set <semantics>Xn</semantics>. We consider the open-set polynomial associated with the topology <semantics>τ</semantics>. Its coefficients are the cardinalities of sets <semantics>Uj=Uj(τ)</semantics> of open sets of size <semantics>j=0,<mspace width="0.277778em"></mspace>…,<mspace width="0.277778em"></mspace>n.</semantics> We prove that this polynomial has only real zeros only in the trivial case where <semantics>τ</semantics> is the discrete topology. Hence, we answer a question raised by J. Brown. We give a partial answer to the question: for which topology is this polynomial log-concave, or at least unimodal? More specifically, we prove that if the topology has a large number of open sets, its open polynomial is unimodal. The idea of degree of log-concavity is introduced and it is shown to be limited for polynomials of non-trivial topologies. Furthermore, the maximum-sized topologies that omit open sets of given sizes are derived. Moreover, all topologies over n points with at least <semantics>(3/8)2n</semantics> open sets are proved to be unimodal, completing previous results.