Time-dependent polynomials with one multiple root and new solvable dynamical systems

A time-dependent monic polynomial in the z variable with N distinct roots such that exactly one root has multiplicity m>=2 is considered. For k=1,2, the k-th derivatives of the N roots are expressed in terms of the derivatives of order j<= k of the first N coefficients of the polynomial and of the derivatives of order j<= k-1 of the roots themselves. These relations are utilized to construct new classes of algebraically solvable first order systems of ODEs as well as N-body problems. Multiple examples of solvable isochronous (all solutions are periodic with the same period) 2- and 3-body problems are provided.