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. For the proof, see the papers. Cohn's theorem is useful to study the distribution of the roots of self-inversive and self-reciprocal polynomials in the complex plane – see, for example, the papers.

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,

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

where

$p^{*}(z)=z^{n}{\bar {p}}\left({\frac {1}{z}}\right)={\bar {p}}_{n}+{\bar {p}}_{n-1}z+\cdots +{\bar {p}}_{0}z^{n}$

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 necessarely self-inversive. The coefficients of a self-inversive polynomial satisfy the relations

$p_{k}=\omega {\bar {p}}_{n-k},\qquad 0\leqslant k\leqslant n.$

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

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

$q(z)=p'(z)=p_{1}+2p_{2}z+\cdots +np_{n}z^{n-1}.$

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

$q^{*}(z)=z^{n-1}{\bar {q}}_{n-1}\left({\frac {1}{z}}\right)=z^{n-1}{\bar {p}}'\left({\frac {1}{z}}\right)=n{\bar {p}}_{n}+(n-1){\bar {p}}_{n-1}z+\cdots +{\bar {p}}_{1}z^{n-1}$

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