  On the one hand, we prove that almost surely, for large dimension, there is
no eigenvalue of a Hermitian polynomial in independent Wigner and deterministic
matrices, in any interval lying at some distance from the supports of a
sequence of deterministic probability measures, which is computed with the
tools of free probability. On the other hand, we establish the strong
asymptotic freeness of independent Wigner matrices and any family of
deterministic matrices with strong limiting distribution.
