  We present a general theory for semiconductor polarons in the framework of
the Froehlich interaction between electrons and phonons. The latter is
investigated using non-commuting phonon creation/annihilation operators
associated with a natural set of non-orthogonal modes. This setting proves
effective for mathematical simplification and physical interpretation and
reveals a nested coupling structure of the Froehlich interaction. The theory is
non-perturbative and well adapted for strong electron-phonon coupling, such as
found in quantum dot (QD) structures. For those particular structures we
introduce a minimal model that allows the computation and qualitative
prediction of the spectrum and geometry of polarons. The model uses a generic
non-orthogonal polaron basis, baptized the "natural basis". Accidental and
symmetry-related electronic degeneracies are studied in detail and are shown to
generate unentangled zero-shift polarons, which we consistently eliminate. As a
practical example, these developments are applied to realistic pyramidal GaAs
QDs. The energy spectrum and the 3D-geometry of polarons are computed and
analyzed, and prove that realistic pyramidal QDs clearly fall in the regime of
strong coupling. Further investigation reveals an unexpected substructure of
"weakly coupled strong coupling regimes", a concept originating from overlap
considerations. Using Bennett's entanglement measure, we finally propose a
heuristic quantification of the coupling strength in QDs.
