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

Symmetry Integrability and Geometry-Methods and Applications

article

Pieter Roffelsen¹ 

[1] Radboud Universiteit Nijmegen

关键词: second Painlev´e equation;    rational solutions;    real roots;    interlacing of roots;    Yablonskii–Vorob’ev polynomials;   

DOI  :  10.3842/SIGMA.2012.099

来源: National Academy of Science of Ukraine

【 摘 要 】

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 n th 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 n th Yablonskii-Vorob'ev polynomial.

【 授权许可】

Unknown