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Cohn's theorem
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In mathematics, Cohn's theorem[1] states that a nth-degree self-inversive polynomial has as many roots in the open unit disk as the reciprocal polynomial of its derivative.[1][2][3] Cohn's theorem is useful for studying the distribution of the roots of self-inversive and self-reciprocal polynomials in the complex plane.[4][5]
An nth-degree polynomial,
is called self-inversive if there exists a fixed complex number ( ) of modulus 1 so that,
where
is the reciprocal polynomial associated with and the bar means complex conjugation. Self-inversive polynomials have many interesting properties.[6] 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.
In the case where 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 is a (n − 1)th-degree polynomial given by
Therefore, Cohn's theorem states that both and the polynomial
have the same number of roots in
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