  We consider a homogeneous mixture of bosons and polarized fermions. We find
that long-range and attractive fermion-mediated interactions between bosons
have dramatic effects on the properties of the bosons. We construct the phase
diagram spanned by boson-fermion mass ratio and boson-fermion scattering
parameter. It consists of stable region of mixing and unstable region toward
phase separation. In stable mixing phase, the collective long-wavelength
excitations can either be well-behaved with infinite lifetime or be finite in
lifetime suffered from the Landau damping. We examine the effects of the
induced interaction on the properties of weakly interacting bosons. It turns
out that the induced interaction not only enhances the repulsion between the
bosons against collapse but also enhances the stability of the superfluid state
by suppressing quantum depletion.
