Poles of Painlevé IV Rationals and their Distribution
We study the distribution of singularities (poles and zeros) of rational solutions of the Painlevé IV equation by means of the isomonodromic deformation method. Singularities are expressed in terms of the roots of generalised Hermite Hm,n and generalised Okamoto Qm,n polynomials. We show that roots...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2018 |
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Інститут математики НАН України
2018
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/209462 |
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| Cite this: | Poles of Painlevé IV Rationals and their Distribution / D. Masoero, P. Roffelsen // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 57 назв. — англ. |
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Masoero, D. Roffelsen, P. 2025-11-21T19:15:33Z 2018 Poles of Painlevé IV Rationals and their Distribution / D. Masoero, P. Roffelsen // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 57 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 34M55; 34M56; 34M60; 33C15; 30C15 arXiv: 1707.05222 https://nasplib.isofts.kiev.ua/handle/123456789/209462 https://doi.org/10.3842/SIGMA.2018.002 We study the distribution of singularities (poles and zeros) of rational solutions of the Painlevé IV equation by means of the isomonodromic deformation method. Singularities are expressed in terms of the roots of generalised Hermite Hm,n and generalised Okamoto Qm,n polynomials. We show that roots of generalised Hermite and Okamoto polynomials are described by an inverse monodromy problem for an anharmonic oscillator of degree two. As a consequence, they turn out to be classified by the monodromy representation of a class of meromorphic functions with a finite number of singularities introduced by Nevanlinna. We compute the asymptotic distribution of roots of the generalized Hermite polynomials in the asymptotic regime when m is large and n fixed. D.M. is an FCT Researcher supported by the FCT Investigator Grant IF/00069/2015. D.M. is also partially supported by the FCT Research Project PTDC/MAT-STA/0975/2014. The present work began in December 2015 while D.M. was a Visiting Scholar at the University of Sydney, funded by the ARC Discovery Project DP130100967. D.M. wishes to thank the Department of Mathematics and Statistics of the University of Sydney and the Centro di Ricerca Matematica Ennio De Giorgi in Pisa for the kind hospitality. P.R. is a research associate at the University of Sydney, supported by Nalini Joshi’s ARC Laureate Fellowship Project FL120100094. P.R. would like to extend his gratitude to the Department of Mathematics of the University of Lisbon and the Centro di Ricerca Matematica Ennio De Giorgi in Pisa, where a major part of this collaboration took place. P.R. was also supported by an IPRS scholarship at the University of Sydney. We are deeply indebted to Nalini Joshi for her continuous scientific and material support. We also thank Alexandre Eremenko, Davide Guzzetti, Peter Miller, and Walter Van Assche for discussions about the present topic of investigation at various stages of this work. We also acknowledge the anonymous referees for helping us improve the paper. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Poles of Painlevé IV Rationals and their Distribution Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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| title |
Poles of Painlevé IV Rationals and their Distribution |
| spellingShingle |
Poles of Painlevé IV Rationals and their Distribution Masoero, D. Roffelsen, P. |
| title_short |
Poles of Painlevé IV Rationals and their Distribution |
| title_full |
Poles of Painlevé IV Rationals and their Distribution |
| title_fullStr |
Poles of Painlevé IV Rationals and their Distribution |
| title_full_unstemmed |
Poles of Painlevé IV Rationals and their Distribution |
| title_sort |
poles of painlevé iv rationals and their distribution |
| author |
Masoero, D. Roffelsen, P. |
| author_facet |
Masoero, D. Roffelsen, P. |
| publishDate |
2018 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
| publisher |
Інститут математики НАН України |
| format |
Article |
| description |
We study the distribution of singularities (poles and zeros) of rational solutions of the Painlevé IV equation by means of the isomonodromic deformation method. Singularities are expressed in terms of the roots of generalised Hermite Hm,n and generalised Okamoto Qm,n polynomials. We show that roots of generalised Hermite and Okamoto polynomials are described by an inverse monodromy problem for an anharmonic oscillator of degree two. As a consequence, they turn out to be classified by the monodromy representation of a class of meromorphic functions with a finite number of singularities introduced by Nevanlinna. We compute the asymptotic distribution of roots of the generalized Hermite polynomials in the asymptotic regime when m is large and n fixed.
|
| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/209462 |
| citation_txt |
Poles of Painlevé IV Rationals and their Distribution / D. Masoero, P. Roffelsen // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 57 назв. — англ. |
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2025-11-29T11:13:52Z |
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2025-11-29T11:13:52Z |
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1850885966969962496 |