Cohn's theorem explained

p(z) has as many roots in the open unit disk D = \{z \in C : |z|< 1 \} as the reciprocal polynomial of its derivative.^{[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,

p(z) = p _{0} + p _{1} z + … + p _{n} z ^{n}

is called self-inversive if there exists a fixed complex number (

\omega ) of modulus 1 so that,p(z) = \omega p ^{*(z),} \left(|\omega| =1 \right),

where

p ^{*(z)=z} ^{n} \bar{p}\left( 1 / \bar{z}\right) = \bar{p} _{n} + \bar{p} _{n-1} z + … + \bar{p} _{0} z ^{n}

is the reciprocal polynomial associated with

p(z) 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.p _{k} = \omega \bar{p} _{n-k} , 0 \leqslant k \leqslant n.

In the case where

\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) is a (n − 1)th-degree polynomial given byq(z) = p'(z) = p _{1} + 2 p _{2} z + … + n p _{n} z ^{n-1} .

Therefore, Cohn's theorem states that both

p(z) and the polynomialq ^{*(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 + … + \bar{p} _{1} z ^{n-1}

have the same number of roots in

|z|< 1 .

See also

• Geometrical properties of polynomial roots

Notes and References

1. Cohn. A. 1922. Über die Anzahl der Wurzeln einer algebraischen Gleichung in einem Kreise. Math. Z.. 14. 110–148. 10.1007/BF01216772.
2. Bonsall. F. F.. Marden. Morris. 1952. Zeros of self-inversive polynomials. Proceedings of the American Mathematical Society. en-US. 3. 3. 471–475. 10.1090/s0002-9939-1952-0047828-8. 0002-9939. 2031905. free.
3. Ancochea. Germán. 1953. Zeros of self-inversive polynomials. Proceedings of the American Mathematical Society. en-US. 4. 6. 900–902. 10.1090/s0002-9939-1953-0058748-8. 0002-9939. 2031826. free.
4. Schinzel. A.. 2005-03-01. Self-Inversive Polynomials with All Zeros on the Unit Circle. The Ramanujan Journal. en. 9. 1–2. 19–23. 10.1007/s11139-005-0821-9. 1382-4090.
5. Vieira. R. S.. 2017. On the number of roots of self-inversive polynomials on the complex unit circle. The Ramanujan Journal. en. 42. 2. 363–369. 10.1007/s11139-016-9804-2. 1382-4090. 1504.00615.
6. Book: Marden, Morris. Geometry of polynomials (revised edition). American Mathematical Society. 1970. 978-0821815038. Mathematical Surveys and Monographs (Book 3) United States of America.