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

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 $\left[\frac{n+1}{2}\right]$. 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.