  For polynomials in independent Wigner matrices, we prove convergence of the
largest singular value to the operator norm of the corresponding polynomial in
free semicircular variables, under fourth moment hypotheses. We actually prove
a more general result of the form "no eigenvalues outside the support of the
limiting eigenvalue distribution." We build on ideas of
Haagerup-Schultz-Thorbj{\o}rnsen on the one hand and Bai-Silverstein on the
other. We refine the linearization trick so as to preserve self-adjointness and
we develop a secondary trick bearing on the calculation of correction terms.
Instead of Poincar\'{e}-type inequalities, we use a variety of matrix
identities and $L^p$ estimates. The Schwinger-Dyson equation controls much of
the analysis.
