  Consider a tuple $(Y_1,\dots,Y_d)$ of normal operators in a tracial operator
algebra setting with prescribed sizes of the eigenspaces for each $Y_i$. We
address the question what one can say about the sizes of the eigenspaces for
any non-commutative polynomial $P(Y_1,\dots,Y_d)$ in those operators? We show
that for each polynomial $P$ there are unavoidable eigenspaces, which occur in
$P(Y_1,\dots,Y_d)$ for any $(Y_1,\dots,Y_d)$ with the prescribed eigenspaces
for the marginals. We will describe this minimal situation both in algebraic
terms - where it is given by realizations via matrices over the free skew field
and via rank calculations - and in analytic terms - where it is given by freely
independent random variables with prescribed atoms in their distributions. The
fact that the latter situation corresponds to this minimal situation allows to
draw many new conclusions about atoms in polynomials of free variables. In
particular, we give a complete description of atoms in the free commutator and
the free anti-commutator. Furthermore, our results do not only apply to
polynomials, but much more general also to non-commutative rational functions.
Since many random matrix models become asymptotically free in the large $N$
limit, our results allow us to calculate the location and size of atoms in the
asymptotic eigenvalue distribution of polynomials and rational functions in
randomly rotated matrices.
