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

Abstract

We study the real roots of the Yablonskii-Vorob'ev polynomials, which are special polynomials used to represent rational solutions of the second Painlev\'e 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.