Jacobian nullwerte and algebraic equations

We present two applications of jacobian nullwerte, both related with the resolution of algebraic equations of any degree. We give a very simple expression of the roots of a polynomial of arbitrary degree in terms of derivatives of hyperelliptic theta functions. This expression can be understood as an explicit proof of Torelliʹs theorem in the hyperelliptic case. We also give geometrical expressions of the discriminant of a polynomial. Both applications are based on a jacobian version of Thomaeʹs formula.