  In polar semiconductors and insulators, the Fr\"ohlich interaction between
electrons and long-wavelength longitudinal optical phonons induces a many-body
renormalization of the carrier effective masses and the appearence of
characteristic phonon sidebands in the spectral function, commonly dubbed
'polaron satellites'. The simplest model that captures these effects is the
Fr\"ohlich model, whereby electrons in a parabolic band interact with a
dispersionless longitudinal optical phonon. The Fr\"ohlich model has been
employed in a number of seminal papers, from early perturbation-theory
approaches to modern diagrammatic Monte Carlo calculations. One limitation of
this model is that it focuses on undoped systems, thus ignoring carrier
screening and Pauli blocking effects that are present in real experiments on
doped samples. To overcome this limitation, we here extend the Fr\"ohlich model
to the case of doped systems, and we provide exact solutions for the electron
spectral function, mass enhancement, and polaron satellites. We perform the
analysis using two approaches, namely Dyson's equation with the Fan-Migdal
self-energy, and the second-order cumulant expansion. We find that these two
approaches provide qualitatively different results. In particular, the Dyson's
approach yields better quasiparticle masses and worse satellites, while the
cumulant approach provides better satellite structures, at the price of worse
quasiparticle masses. Both approaches yield an anomalous enhancement of the
electron effective mass at finite doping levels, which in turn leads to a
breakdown of the quasiparticle picture in a significant portion of the phase
diagram.
