A Constructive Proof for the Umemura Polynomials of the Third Painlevé Equation
We are concerned with the Umemura polynomials associated with rational solutions of the third Painlevé equation. We extend Taneda's method, which was developed for the Yablonskii-Vorob'ev polynomials associated with the second Painlevé equation, to give an algebraic proof that the rational...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2023 |
| Автори: | , , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут математики НАН України
2023
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/212004 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | A Constructive Proof for the Umemura Polynomials of the Third Painlevé Equation. Peter A. Clarkson, Chun-Kong Law and Chia-Hua Lin. SIGMA 19 (2023), 080, 20 pages |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862742197397356544 |
|---|---|
| author | Clarkson, Peter A. Law, Chun-Kong Lin, Chia-Hua |
| author_facet | Clarkson, Peter A. Law, Chun-Kong Lin, Chia-Hua |
| citation_txt | A Constructive Proof for the Umemura Polynomials of the Third Painlevé Equation. Peter A. Clarkson, Chun-Kong Law and Chia-Hua Lin. SIGMA 19 (2023), 080, 20 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We are concerned with the Umemura polynomials associated with rational solutions of the third Painlevé equation. We extend Taneda's method, which was developed for the Yablonskii-Vorob'ev polynomials associated with the second Painlevé equation, to give an algebraic proof that the rational functions generated by the nonlinear recurrence relation that determines the Umemura polynomials are indeed polynomials. Our proof is constructive and gives information about the roots of the Umemura polynomials.
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| first_indexed | 2026-04-17T18:09:51Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-212004 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-04-17T18:09:51Z |
| publishDate | 2023 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Clarkson, Peter A. Law, Chun-Kong Lin, Chia-Hua 2026-01-22T09:14:12Z 2023 A Constructive Proof for the Umemura Polynomials of the Third Painlevé Equation. Peter A. Clarkson, Chun-Kong Law and Chia-Hua Lin. SIGMA 19 (2023), 080, 20 pages 1815-0659 2020 Mathematics Subject Classification: 33E17; 34M55; 65Q30 arXiv:1609.00495 https://nasplib.isofts.kiev.ua/handle/123456789/212004 https://doi.org/10.3842/SIGMA.2023.080 We are concerned with the Umemura polynomials associated with rational solutions of the third Painlevé equation. We extend Taneda's method, which was developed for the Yablonskii-Vorob'ev polynomials associated with the second Painlevé equation, to give an algebraic proof that the rational functions generated by the nonlinear recurrence relation that determines the Umemura polynomials are indeed polynomials. Our proof is constructive and gives information about the roots of the Umemura polynomials. The authors are deeply indebted to the anonymous referees for their very careful and constructive review. PAC thanks Thomas Bothner, Alfredo Deano, Clare Dunning, Marco Fasondini, Kerstin Jordaan, Ana Loureiro and Walter Van Assche for their helpful comments and illuminating discussions and also the Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung, Taiwan and the Department of Mathematics, National Taiwan University, Taipei, Taiwan, for their hospitality during his visit where some of this work was done. CKL also thanks Peter Miller, Yik-Man Chiang, and Guofu Yu for stimulating discussions. CKL is partially supported by the National Science and Technology Council (formerly Ministry of Science and Technology), Taiwan. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications A Constructive Proof for the Umemura Polynomials of the Third Painlevé Equation Article published earlier |
| spellingShingle | A Constructive Proof for the Umemura Polynomials of the Third Painlevé Equation Clarkson, Peter A. Law, Chun-Kong Lin, Chia-Hua |
| title | A Constructive Proof for the Umemura Polynomials of the Third Painlevé Equation |
| title_full | A Constructive Proof for the Umemura Polynomials of the Third Painlevé Equation |
| title_fullStr | A Constructive Proof for the Umemura Polynomials of the Third Painlevé Equation |
| title_full_unstemmed | A Constructive Proof for the Umemura Polynomials of the Third Painlevé Equation |
| title_short | A Constructive Proof for the Umemura Polynomials of the Third Painlevé Equation |
| title_sort | constructive proof for the umemura polynomials of the third painlevé equation |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/212004 |
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