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Cohn's theorem
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<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, Cohn's theorem states that a <i>n</i>th-degree self-inversive <a href="/facts/Polynomial/Lzak8VVx">polynomial</a> p ( z ) {\displaystyle p(z)} has as many <a href="/facts/Zeroes_of_a_function/mlK3dhoT">roots</a> in the open unit disk D = { z ∈ C : | z | < 1 } {\displaystyle D=\{z\in \mathbb {C} :|z|<1\}} as the <a href="/facts/Reciprocal_polynomial/RSXNJKv8">reciprocal polynomial</a> of its <a href="/facts/Derivative/7Ito70Dh">derivative</a>. Cohn's theorem is useful for studying the distribution of the roots of self-inversive and self-reciprocal polynomials in the <a href="/facts/Complex_plane/vE0QrxJp">complex plane</a>. </p><p>An <i>n</i>th-degree polynomial, </p> p ( z ) = p 0 + p 1 z + ⋯ + p n z n {\displaystyle p(z)=p_{0}+p_{1}z+\cdots +p_{n}z^{n}} <p>is called self-inversive if there exists a <i>fixed</i> <a href="/facts/Complex_number/2w7mqI7e">complex number</a> ( ω {\displaystyle \omega } ) of <a href="/facts/Modular_arithmetic/0wtRa5dU">modulus</a> 1 so that, </p> p ( z ) = ω p ∗ ( z ) , ( | ω | = 1 ) , {\displaystyle p(z)=\omega p^{*}(z),\qquad \left(|\omega |=1\right),} <p>where </p> 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}} <p>is the <a href="/facts/Reciprocal_polynomial/RSXNJKv8">reciprocal polynomial</a> associated with p ( z ) {\displaystyle p(z)} and the bar means <a href="/facts/Complex_conjugate/7TEaoQDe">complex conjugation</a>. Self-inversive polynomials have many interesting properties. For instance, its roots are all <a href="/facts/Symmetric_polynomial/GNhguVcn">symmetric</a> with respect to the <a href="/facts/Unit_circle/VsTffjPK">unit circle</a> and a polynomial whose roots are all on the unit circle is necessarily self-inversive. The <a href="/facts/Coefficient/5Hwq2Wuq">coefficients</a> of self-inversive polynomials satisfy the relations. </p> p k = ω p ¯ n − k , 0 ⩽ k ⩽ n . {\displaystyle p_{k}=\omega {\bar {p}}_{n-k},\qquad 0\leqslant k\leqslant n.} <p>In the case where ω = 1 , {\displaystyle \omega =1,} a <i>self-inversive polynomial</i> becomes a <i>complex-reciprocal polynomial</i> (also known as a <i>self-conjugate polynomial</i>). If its coefficients are real then it becomes a <i>real self-reciprocal polynomial</i>. </p><p>The <i><a href="/facts/Formal_derivative/actYz19a">formal derivative</a></i> of p ( z ) {\displaystyle p(z)} is a (<i>n</i> − 1)th-degree polynomial given by </p> 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}.} <p>Therefore, Cohn's theorem states that both p ( z ) {\displaystyle p(z)} and the polynomial </p> 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}} <p>have the same number of roots in | z | < 1. {\displaystyle |z|<1.} </p>