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...
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2025-01-01
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| author | Moussa Benoumhani Brahim Chaourar |
| author_facet | Moussa Benoumhani Brahim Chaourar |
| author_sort | Moussa Benoumhani |
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| description | 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. |
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| spelling | doaj-art-2ba9b922c1754e61b98f86a66b3bafd02025-08-20T03:11:58ZengMDPI AGAxioms2075-16802025-01-0114210310.3390/axioms14020103On Polynomials Associated with Finite TopologiesMoussa Benoumhani0Brahim Chaourar1Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11586, Saudi ArabiaDepartment of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11586, Saudi ArabiaLet <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.https://www.mdpi.com/2075-1680/14/2/103finite topologylog-concavepolynomialunimodalzeros |
| spellingShingle | Moussa Benoumhani Brahim Chaourar On Polynomials Associated with Finite Topologies Axioms finite topology log-concave polynomial unimodal zeros |
| title | On Polynomials Associated with Finite Topologies |
| title_full | On Polynomials Associated with Finite Topologies |
| title_fullStr | On Polynomials Associated with Finite Topologies |
| title_full_unstemmed | On Polynomials Associated with Finite Topologies |
| title_short | On Polynomials Associated with Finite Topologies |
| title_sort | on polynomials associated with finite topologies |
| topic | finite topology log-concave polynomial unimodal zeros |
| url | https://www.mdpi.com/2075-1680/14/2/103 |
| work_keys_str_mv | AT moussabenoumhani onpolynomialsassociatedwithfinitetopologies AT brahimchaourar onpolynomialsassociatedwithfinitetopologies |