Let $P(z)$ be a  polynomial with complex variable $z$.  We consider the following distribution for the roots of  $P(z)=0$: the distribution is a triple $(n_{1},n_{2},n_{3})$
where these integers are the number of roots in the interior of unite circle, on the unit circle and out of unit disc, respectively.

$\ell^{2}$ is the Hilbert space of all square sumable sequence of  complex numbers. $S_1$ is the shift operator on $\ell^{2}.$

**The equivalence relation:** We say two polynomials $P,Q$  are equivalent if $P(S_1)$ is  conjugate to $Q(S_1)$  via  an invertible operator in $B(\ell^{2})$.

Assume that $P,Q$ are two equivalent polynomials:

>1.Must they have the same degree?
>
>2.Must they have the same root distribution?


The motivation for the second question is that when $P,Q$ have the same degree and $P$ has $(n_{1},0,n_{3})$ distribution, then $Q$ has the same distribution as $P$. The reason is that $P(S_1)$ is  a  Fredhom operator of index $-n_1$. Obviously this property is invariant under conjucacy. So $Q$ has the same root  distribution.