Английская Википедия:Cohn's theorem

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In mathematics, Cohn's theorem states that a nth-degree self-inversive polynomial <math>p(z)</math> has as many roots in the open unit disk <math>D =\{z \in \mathbb{C}: |z|<1\}</math> 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,

<math>p(z) = p_0 + p_1 z + \cdots + p_n z^n </math>

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

<math>p(z) = \omega p^*(z),\qquad \left(|\omega|=1\right),</math>

where

<math>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</math>

is the reciprocal polynomial associated with <math>p(z)</math> 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.

<math>p_k = \omega \bar{p}_{n-k}, \qquad 0 \leqslant k \leqslant n. </math>

In the case where <math>\omega = 1, </math> 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 <math>p(z)</math> is a (n − 1)th-degree polynomial given by

<math>q(z) =p'(z) = p_1 + 2p_2 z + \cdots + n p_n z^{n-1}. </math>

Therefore, Cohn's theorem states that both <math>p(z)</math> and the polynomial

<math>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} </math>

have the same number of roots in <math>|z|<1.</math>

See also

• Geometrical properties of polynomial roots