Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 8 (2012), 099, 9 pages      arXiv:1208.2337      https://doi.org/10.3842/SIGMA.2012.099

On the Number of Real Roots of the Yablonskii-Vorob'ev Polynomials

Pieter Roffelsen
Radboud Universiteit Nijmegen, IMAPP, FNWI, Heyendaalseweg 135, 6525 AJ Nijmegen, The Netherlands

Received August 14, 2012, in final form December 07, 2012; Published online December 14, 2012

Abstract
We study the real roots of the Yablonskii-Vorob'ev polynomials, which are special polynomials used to represent rational solutions of the second Painlevé equation. It has been conjectured that the number of real roots of the nth Yablonskii-Vorob'ev polynomial equals [(n+1)/2]. We prove this conjecture using an interlacing property between the roots of the Yablonskii-Vorob'ev polynomials. Furthermore we determine precisely the number of negative and the number of positive real roots of the nth Yablonskii-Vorob'ev polynomial.

Key words: second Painlevé equation; rational solutions; real roots; interlacing of roots; Yablonskii-Vorob'ev polynomials.

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