Discriminants of Umemura polynomials associated to Painlevé III

Explicit discriminant and resultant formulas for the Umemura polynomials associated with rational solutions to Painlevé III equation are proved; P.A. Clarkson conjectured these results, and the formulas are expressed as products of small integers to large powers, and the discriminants can also be used to determine when the roots of Umemura's special polynomials, which have very interesting structure and striking symmetry, coalesce in the complex plane.