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

TL;DR

This paper proves a conjecture about the number of real roots of Yablonskii-Vorob'ev polynomials, showing it equals [(n+1)/2], and precisely counts positive and negative roots.

Contribution

It establishes the exact count of real roots of Yablonskii-Vorob'ev polynomials and proves the conjectured root count using an interlacing property.

Findings

Number of real roots equals [(n+1)/2]

Exact counts of positive and negative roots

Interlacing property of roots

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.