Cohn's theorem

In mathematics, Cohn's theorem{{Cite journal|last=Cohn|first=A|date=1922|title=Über die Anzahl der Wurzeln einer algebraischen Gleichung in einem Kreise|journal=Math. Z.|volume=14|pages=110–148|doi=10.1007/BF01216772}} states that a nth-degree self-inversive polynomial $p(z)$ has as many roots in the open unit disk as the reciprocal polynomial of its derivative.{{Cite journal|last=Bonsall|first=F. F.|last2=Marden|first2=Morris|date=1952|title=Zeros of self-inversive polynomials|url=https://www.ams.org/home/page/|journal=Proceedings of the American Mathematical Society|language=en-US|volume=3|issue=3|pages=471–475|doi=10.1090/s0002-9939-1952-0047828-8|issn=0002-9939|jstor=2031905|doi-access=free}}{{Cite journal|last=Ancochea|first=Germán|date=1953|title=Zeros of self-inversive polynomials|url=https://www.ams.org/home/page/|journal=Proceedings of the American Mathematical Society|language=en-US|volume=4|issue=6|pages=900–902|doi=10.1090/s0002-9939-1953-0058748-8|issn=0002-9939|jstor=2031826|doi-access=free}} Cohn's theorem is useful for studying the distribution of the roots of self-inversive and self-reciprocal polynomials in the complex plane.{{Cite journal|last=Schinzel|first=A.|date=2005-03-01|title=Self-Inversive Polynomials with All Zeros on the Unit Circle|journal=The Ramanujan Journal|language=en|volume=9|issue=1–2|pages=19–23|doi=10.1007/s11139-005-0821-9|issn=1382-4090}}{{Cite journal|last=Vieira|first=R. S.|date=2017|title=On the number of roots of self-inversive polynomials on the complex unit circle|journal=The Ramanujan Journal|language=en|volume=42|issue=2|pages=363–369|doi=10.1007/s11139-016-9804-2|issn=1382-4090|arxiv=1504.00615}}

An nth-degree polynomial,

: $p(z)\; =\; p\_0\; +\; p\_1\; z\; +\; \backslash cdots\; +\; p\_n\; z^n$

is called self-inversive if there exists a fixed complex number ( $\backslash omega$ ) of modulus 1 so that,

: $p(z)\; =\; \backslash omega\; p^*(z),\backslash qquad\; \backslash left(|\backslash omega|=1\backslash right),$

where

: $p^*(z)=z^n\; \backslash bar\{p\}\backslash left(1\; /\; \backslash bar\{z\}\backslash right)\; =\backslash bar\{p\}\_n\; +\; \backslash bar\{p\}\_\{n-1\}\; z\; +\; \backslash cdots\; +\; \backslash bar\{p\}\_0\; z^n$

is the reciprocal polynomial associated with $p(z)$ and the bar means complex conjugation. Self-inversive polynomials have many interesting properties.{{Cite book|title=Geometry of polynomials (revised edition)|last=Marden|first=Morris|publisher=American Mathematical Society|year=1970|isbn=978-0821815038|location=Mathematical Surveys and Monographs (Book 3) United States of America|pages=}} 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.

: $p\_k\; =\; \backslash omega\; \backslash bar\{p\}\_\{n-k\},\; \backslash qquad\; 0\; \backslash leqslant\; k\; \backslash leqslant\; n.$

In the case where $\backslash omega\; =\; 1,$ 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 by

: $q(z)\; =p\text{'}(z)\; =\; p\_1\; +\; 2p\_2\; z\; +\; \backslash cdots\; +\; n\; p\_n\; z^\{n-1\}.$

Therefore, Cohn's theorem states that both $p(z)$ and the polynomial

: $q^*(z)\; =z^\{n-1\}\backslash bar\{q\}\_\{n-1\}\backslash left(1\; /\; \backslash bar\{z\}\backslash right)\; =\; z^\{n-1\}\; \backslash bar\{p\}\text{'}\; \backslash left(1\; /\; \backslash bar\{z\}\backslash right)\; =\; n\; \backslash bar\{p\}\_n\; +\; (n-1)\backslash bar\{p\}\_\{n-1\}\; z\; +\; \backslash cdots\; +\; \backslash bar\{p\}\_1\; z^\{n-1\}$

have the same number of roots in