Discriminants of Some Painlevé Polynomials

In this paper, we derive explicit formulas for the discriminants of the Yablonsky-Vorobiev polynomials P_m (x), the bi-Hermite polynomials H_m,n (x) and the Okamoto polynomials Q_m,n (x), as well as some related resultants. In all three cases, the discriminant and related resultants factor into a product of small primes only. Our formula in the bi-Hermite case reduces when https://www.w3.org/1998/Math/MathML"> n = 1 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780138747022/abb99e87-ffb7-4196-a4c4-acb0033e3d6a/content/eq2410.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> to the nineteenth-century formula for the discriminant of a Hermite polynomial. Our other two discriminant formulas do not have direct antecedents. The introduction and final section provide some context for our results.