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A time-dependent monic polynomial in the z variable with N distinct rootssuch that exactly one root has multiplicity m>=2 is considered. For k=1,2, thek-th derivatives of the N roots are expressed in terms of the derivatives oforder j<= k of the first N coefficients of the polynomial and of thederivatives of order j<= k-1 of the roots themselves. These relations areutilized to construct new classes of algebraically solvable first order systemsof 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 areprovided.

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