In mathematics, Cohn's theorem states that a nth-degree self-inversive polynomial p ( z ) {\displaystyle p(z)} has as many roots in the open unit disk D = { z ∈ C : | z | < 1 } {\displaystyle 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 + ⋯ + p n z n {\displaystyle p(z)=p_{0}+p_{1}z+\cdots +p_{n}z^{n}}

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

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

where

p ∗ ( z ) = z n p ¯ ( 1 / z ¯ ) = p ¯ n + p ¯ n − 1 z + ⋯ + p ¯ 0 z n {\displaystyle p^{*}(z)=z^{n}{\bar {p}}\left(1/{\bar {z}}\right)={\bar {p}}_{n}+{\bar {p}}_{n-1}z+\cdots +{\bar {p}}_{0}z^{n}}

is the reciprocal polynomial associated with p ( z ) {\displaystyle 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 = ω p ¯ n − k , 0 ⩽ k ⩽ n . {\displaystyle p_{k}=\omega {\bar {p}}_{n-k},\qquad 0\leqslant k\leqslant n.}

In the case where ω = 1 , {\displaystyle \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 ) {\displaystyle p(z)} is a (n − 1)th-degree polynomial given by

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

Therefore, Cohn's theorem states that both p ( z ) {\displaystyle p(z)} and the polynomial

q ∗ ( z ) = z n − 1 q ¯ n − 1 ( 1 / z ¯ ) = z n − 1 p ¯ ′ ( 1 / z ¯ ) = n p ¯ n + ( n − 1 ) p ¯ n − 1 z + ⋯ + p ¯ 1 z n − 1 {\displaystyle q^{*}(z)=z^{n-1}{\bar {q}}_{n-1}\left(1/{\bar {z}}\right)=z^{n-1}{\bar {p}}'\left(1/{\bar {z}}\right)=n{\bar {p}}_{n}+(n-1){\bar {p}}_{n-1}z+\cdots +{\bar {p}}_{1}z^{n-1}}

have the same number of roots in | z | < 1. {\displaystyle |z|<1.}