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 inside 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 equivalent relation: We say two poynomials $P,Q$ 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 resaon 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.