有限拓扑的开多项式
The open polynomials of the finite topologies
摘要：
设T是有限集Xn上的拓扑.我们考虑与拓扑T相关联的开多项式。它的系数是大小为j=0，...，n. J. Brown [4]问这个多项式何时只有实零点。我们证明了这个多项式有实零点，只有在平凡的情况下，T是离散拓扑。然后，我们削弱布朗的问题：这个多项式的拓扑是对数凹的，或至少单峰？给出了部分答案。更确切地说，我们证明了如果拓扑有大量的开集，那么它的开多项式是单峰的。
英文摘要：
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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