Cohn's theorem
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In mathematics, Cohn's theorem[1] states that a nth-degree self-inversive polynomial [math]\displaystyle{ p(z) }[/math] has as many roots in the open unit disk [math]\displaystyle{ D =\{z \in \mathbb{C}: |z|\lt 1\} }[/math] 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,
- [math]\displaystyle{ 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]\displaystyle{ \omega }[/math] ) of modulus 1 so that,
- [math]\displaystyle{ p(z) = \omega p^*(z),\qquad \left(|\omega|=1\right), }[/math]
where
- [math]\displaystyle{ p^*(z)=z^n \bar{p}\left(\frac{1}{z}\right) =\bar{p}_n + \bar{p}_{n-1} z + \cdots + \bar{p}_0 z^n }[/math]
is the reciprocal polynomial associated with [math]\displaystyle{ p(z) }[/math] 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.
- [math]\displaystyle{ p_k = \omega \bar{p}_{n-k}, \qquad 0 \leqslant k \leqslant n. }[/math]
In the case where [math]\displaystyle{ \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]\displaystyle{ p(z) }[/math] is a (n − 1)th-degree polynomial given by
- [math]\displaystyle{ 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]\displaystyle{ p(z) }[/math] as the polynomial
- [math]\displaystyle{ q^*(z) =z^{n-1}\bar{q}_{n-1}\left(\frac{1}{z}\right) = z^{n-1} \bar{p}' \left(\frac{1}{z}\right) = n \bar{p}_n + (n-1)\bar{p}_{n-1} z + \cdots + \bar{p}_1 z^{n-1} }[/math]
has the same number of roots in [math]\displaystyle{ |z|\lt 1. }[/math]
References
- ↑ 1.0 1.1 Cohn, A (1922). "Über die Anzahl der Wurzeln einer algebraischen Gleichung in einem Kreise". Math. Z. 14: 110–148. doi:10.1007/BF01216772.
- ↑ Bonsall, F. F.; Marden, Morris (1952). "Zeros of self-inversive polynomials" (in en-US). Proceedings of the American Mathematical Society 3 (3): 471–475. doi:10.1090/s0002-9939-1952-0047828-8. ISSN 0002-9939. http://www.ams.org/home/page/.
- ↑ Ancochea, Germán (1953). "Zeros of self-inversive polynomials" (in en-US). Proceedings of the American Mathematical Society 4 (6): 900–902. doi:10.1090/s0002-9939-1953-0058748-8. ISSN 0002-9939. http://www.ams.org/home/page/.
- ↑ Schinzel, A. (2005-03-01). "Self-Inversive Polynomials with All Zeros on the Unit Circle" (in en). The Ramanujan Journal 9 (1–2): 19–23. doi:10.1007/s11139-005-0821-9. ISSN 1382-4090.
- ↑ Vieira, R. S. (2017). "On the number of roots of self-inversive polynomials on the complex unit circle" (in en). The Ramanujan Journal 42 (2): 363–369. doi:10.1007/s11139-016-9804-2. ISSN 1382-4090.
- ↑ Marden, Morris (1970). Geometry of polynomials (revised edition). Mathematical Surveys and Monographs (Book 3) United States of America: American Mathematical Society. ISBN 978-0821815038.
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