  We show that almost all the zeros of any finite linear combination of
independent characteristic polynomials of random unitary matrices lie on the
unit circle. This result is the random matrix analogue of an earlier result by
Bombieri and Hejhal on the distribution of zeros of linear combinations of
$L$-functions, thus providing further evidence for the conjectured links
between the value distribution of the characteristic polynomial of random
unitary matrices and the value distribution of $L$-functions on the critical
line.
