Cohn's theorem

In the theory od complex polynomials, Cohn's theorem states that a n-degree self-inversive polynomial $p_{n}(z)$ 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-reciprcal polynomials in the complex plane - see, for example, the papers.

A n-degree polynomial,

$p_{n}(z)=p_{0}+p_{1}z+\cdots +p_{n}z^{n}$

is called self-inversive if

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

where,

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

is the reciprocal polynomial associated with $p_{n}(z)$ 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 $p_{n}(z)$ is a n-1 degree polynomial given by

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

Therefore, Cohn's theorem states that both $p_{n}(z)$ as the polynomial

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

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