  The ground-state energy, the addition energies and the optical absorption
spectra are derived for interacting polarons in parabolic quantum dots in three
and two dimensions. A path integral formalism for identical particles is used
in order to take into account the fermion statistics. The approach is applied
to both closed-shell and open-shell systems of interacting polarons. Using a
generalization of the Jensen-Feynman variational principle, the ground-state
energy of a confined N-polaron system is analyzed as a function of N and of the
electron-phonon coupling constant. As distinct from the few-electron systems
without the electron-phonon interaction, three types of spin polarization are
possible for the ground state of the few-polaron systems: (i) a spin-polarized
state, (ii) a state where the spin is determined by Hund's rule, (iii) a state
with the minimal possible spin. A transition from a state fulfilling Hund's
rule, to a spin-polarized state occurs when decreasing the electron density. In
the strong-coupling limit, the system of interacting polarons turns into a
state with the minimal possible spin. These transitions should be
experimentally observable in the optical absorption spectra of quantum dots.
