Schur Characteristic Polynomials

Schur

A characteristic polynomial $\Delta [z]$ is Schur if $\text{roots}(\Delta )\subset \mathbb{D}$.

Lemma

Let $\Delta [z]={\sum }_{i=0}^n{c}_i{z}^i$. If $\Delta$ is Schur, then $|c_n|>|c_0|$.

Note that this does not go the other way! We could have $|c_n|>|c_0|$ but the polynomial is not Schur.

Example 1:

$$z^3+z+3=\Delta [z]$$

where

• $c_n=c_3=1$
• $c_1=1$
• $c_0=3$

Then:

$$\begin{array}{r}|c_n|=|c_3|=1\\ |c_0|=3\end{array}⟹|c_n|<|c_0|⟹\Delta \text{ is NOT Schur!}$$

Example 2:

$$z^2+\frac{1}{2}z+\frac{1}{2}=\Delta [z]$$

Here, $|c_n|=1$ and $|c_0|=\frac{1}{2}$. Thus, $|c_n|>|c_0|$.

• However, if we factor out the characteristic polynomial, we can see that $\Delta z=(z+2)\left(z+\frac{1}{4}\right)$, so the polynomial is not Schur! (Shows the limitations of the lemma)