Hyperdeterminantal Total Positivity
For a given positive integer , the concept of hyperdeterminantal total positivity is defined for a kernel : ℝ²ᵐ → ℝ, thereby generalizing the classical concept of total positivity. Extending the fundamental example, (, ) = exp(), , ∈ ℝ, of a classical totally positive kernel, the hyperdeterminantal...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2025 |
| Main Authors: | , |
| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2025
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/213521 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Hyperdeterminantal Total Positivity. Kenneth W. Johnson and Donald St. P. Richards. SIGMA 21 (2025), 055, 18 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862717478797312000 |
|---|---|
| author | Johnson, Kenneth W. Richards, Donald St. P. |
| author_facet | Johnson, Kenneth W. Richards, Donald St. P. |
| citation_txt | Hyperdeterminantal Total Positivity. Kenneth W. Johnson and Donald St. P. Richards. SIGMA 21 (2025), 055, 18 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | For a given positive integer , the concept of hyperdeterminantal total positivity is defined for a kernel : ℝ²ᵐ → ℝ, thereby generalizing the classical concept of total positivity. Extending the fundamental example, (, ) = exp(), , ∈ ℝ, of a classical totally positive kernel, the hyperdeterminantal total positivity property of the kernel (₁, …, ₂ₘ) = exp(₁⋯₂ₘ), ₁,…,₂ₘ ∈ ℝ is established. By applying Matsumoto's hyperdeterminantal Binet-Cauchy formula, we derive a generalization of Karlin's basic composition formula. We use the generalized composition formula to construct several examples of hyperdeterminantal totally positive kernels. Further generalizations of hyperdeterminantal total positivity via the theory of finite reflection groups are described, and some open problems are posed.
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| first_indexed | 2026-03-20T16:46:49Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-213521 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-20T16:46:49Z |
| publishDate | 2025 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Johnson, Kenneth W. Richards, Donald St. P. 2026-02-18T11:24:01Z 2025 Hyperdeterminantal Total Positivity. Kenneth W. Johnson and Donald St. P. Richards. SIGMA 21 (2025), 055, 18 pages 1815-0659 2020 Mathematics Subject Classification: 33C20; 05E05; 15A15; 15A72; 33C80 arXiv:2412.03000 https://nasplib.isofts.kiev.ua/handle/123456789/213521 https://doi.org/10.3842/SIGMA.2025.055 For a given positive integer , the concept of hyperdeterminantal total positivity is defined for a kernel : ℝ²ᵐ → ℝ, thereby generalizing the classical concept of total positivity. Extending the fundamental example, (, ) = exp(), , ∈ ℝ, of a classical totally positive kernel, the hyperdeterminantal total positivity property of the kernel (₁, …, ₂ₘ) = exp(₁⋯₂ₘ), ₁,…,₂ₘ ∈ ℝ is established. By applying Matsumoto's hyperdeterminantal Binet-Cauchy formula, we derive a generalization of Karlin's basic composition formula. We use the generalized composition formula to construct several examples of hyperdeterminantal totally positive kernels. Further generalizations of hyperdeterminantal total positivity via the theory of finite reflection groups are described, and some open problems are posed. The results in this article were first presented in [20] at the Second CREST-SBM International Conference, “Harmony of Gröbner Bases and the Modern Industrial Society”, held June 28 July 2, 2010, in Osaka, Japan (see [18]). We express our gratitude to the organizers of the meeting for the opportunity to present our results there. We are also grateful to SIGMA’s referees for their meticulous reading of the article and very helpful comments. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Hyperdeterminantal Total Positivity Article published earlier |
| spellingShingle | Hyperdeterminantal Total Positivity Johnson, Kenneth W. Richards, Donald St. P. |
| title | Hyperdeterminantal Total Positivity |
| title_full | Hyperdeterminantal Total Positivity |
| title_fullStr | Hyperdeterminantal Total Positivity |
| title_full_unstemmed | Hyperdeterminantal Total Positivity |
| title_short | Hyperdeterminantal Total Positivity |
| title_sort | hyperdeterminantal total positivity |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/213521 |
| work_keys_str_mv | AT johnsonkennethw hyperdeterminantaltotalpositivity AT richardsdonaldstp hyperdeterminantaltotalpositivity |