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On the irreducibility conjecture based on computer calculation for Yablonskii-Vorobiev polynomials which give a rational solution of the Toda equation of Painlevé-II type

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Abstract

An interesting sequence of polynomials which were first introduced by A. I. Yablonskii and A. P. Vorobiev to describe rational solutions of the Painlevé-II equation is considered. It also gives a special rational solution of the Toda equation. All our polynomials have integral coefficients. By computer calculation we proved rigorously that the first 24 of our polynomials are irreducible. It is natural to suppose that all our polynomials are irreducible. But our numerical results show that it is a very difficult mathematical problem to determine whether or not all our polynomials are irreducible or not. E. R. Berlekamp’s algorithm was used for numerical calculation instead of usual Kronecker’s algorithm.

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Authors and Affiliations

  1. Department of Applied Mathematics, Faculty of Engineering, Ehime University, 790, Matsuyama, Japan

    Yoshinori Kametaka

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  1. Yoshinori Kametaka
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This paper is dedicated to late Professor T. Miyata.

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Kametaka, Y. On the irreducibility conjecture based on computer calculation for Yablonskii-Vorobiev polynomials which give a rational solution of the Toda equation of Painlevé-II type. Japan J. Appl. Math. 2, 241–246 (1985). https://doi.org/10.1007/BF03167047

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  • DOI: https://doi.org/10.1007/BF03167047

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