Cohn's theorem

In mathematics, Cohn's theorem states that a nth-degree self-inversive polynomial $p(z)$ has as many roots in the open unit disk $D=\{z\in \mathbb{ℂ}:|z|<1\}$ as the reciprocal polynomial of its derivative. Cohn's theorem is useful for studying the distribution of the roots of self-inversive and self-reciprocal polynomials in the complex plane.

An nth-degree polynomial,

$p(z)=p_0+p_{1}z+\cdots +p_n{z}^n$

is called self-inversive if there exists a fixed complex number ( $\omega$ ) of modulus 1 so that,

$p(z)=\omega p^{*}(z),(|\omega |=1),$

where

$p^{*}(z)=z^n\overline{p}\left(\frac{1}{z}\right)={\overline{p}}_n+{\overline{p}}_{n-1}z+\cdots +{\overline{p}}_0{z}^n$

is the reciprocal polynomial associated with $p(z)$ and the bar means complex conjugation. Self-inversive polynomials have many interesting properties. For instance, its roots are all symmetric with respect to the unit circle and a polynomial whose roots are all on the unit circle is necessarily self-inversive. The coefficients of self-inversive polynomials satisfy the relations.

$p_k=\omega {\overline{p}}_{n-k},0⩽k⩽n.$

In the case where $\omega =1,$ a self-inversive polynomial becomes a complex-reciprocal polynomial (also known as a self-conjugate polynomial). If its coefficients are real then it becomes a real self-reciprocal polynomial.

The formal derivative of $p(z)$ is a (n − 1)th-degree polynomial given by

$q(z)=p^{\prime }(z)=p_1+2p_{2}z+\cdots +n{p}_n{z}^{n-1}.$

Therefore, Cohn's theorem states that both $p(z)$ as the polynomial

$q^{*}(z)=z^{n-1}{\overline{q}}_{n-1}\left(\frac{1}{z}\right)=z^{n-1}{\overline{p}}^{\prime }\left(\frac{1}{z}\right)=n{\overline{p}}_n+(n-1){\overline{p}}_{n-1}z+\cdots +{\overline{p}}_1{z}^{n-1}$

has the same number of roots in $|z|<1.$