On Polynomials Associated with Finite Topologies
Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>τ</mi></semantics></math></inline-formula> be a topology on the finite set <inline-formula><math xmlns="http://www.w3...
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2025-01-01
|
| Series: | Axioms |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2075-1680/14/2/103 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Summary: | Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>τ</mi></semantics></math></inline-formula> be a topology on the finite set <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>X</mi><mi>n</mi></msub></semantics></math></inline-formula>. We consider the open-set polynomial associated with the topology <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>τ</mi></semantics></math></inline-formula>. Its coefficients are the cardinalities of sets <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>U</mi><mi>j</mi></msub><mo>=</mo><msub><mi>U</mi><mi>j</mi></msub><mrow><mo>(</mo><mi>τ</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> of open sets of size <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>j</mi><mo>=</mo><mn>0</mn><mo>,</mo><mspace width="0.277778em"></mspace><mo>…</mo><mo>,</mo><mspace width="0.277778em"></mspace><mi>n</mi><mo>.</mo></mrow></semantics></math></inline-formula> We prove that this polynomial has only real zeros only in the trivial case where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>τ</mi></semantics></math></inline-formula> 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 <i>n</i> points with at least <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo>(</mo><mn>3</mn><mo>/</mo><mn>8</mn><mo>)</mo></mrow><msup><mn>2</mn><mi>n</mi></msup></mrow></semantics></math></inline-formula> open sets are proved to be unimodal, completing previous results. |
|---|---|
| ISSN: | 2075-1680 |