A polynomial can have multiple zeros.
If is a d-th root of unity,
then is a zero of
and therefore a zero of for any k such that d | k-1.
Hence it is a zero of multiplicity 2 for , a polynomial in P.
Higher multiplicities can be obtained by iterating this procedure.
On the other hand, we do not know whether any that is not a root of unity can be a multiple root of any .
There do exist power series with coefficients 0,1 that have double
zeros z with |z| <1,
as will be shown in Section 3.