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Cohn's theorem

From Wikipedia, the free encyclopedia

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

See also

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References

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  1. ^ a b Cohn, A (1922). "Über die Anzahl der Wurzeln einer algebraischen Gleichung in einem Kreise". Math. Z. 14: 110–148. doi:10.1007/BF01216772.
  2. ^ Bonsall, F. F.; Marden, Morris (1952). "Zeros of self-inversive polynomials". Proceedings of the American Mathematical Society. 3 (3): 471–475. doi:10.1090/s0002-9939-1952-0047828-8. ISSN 0002-9939. JSTOR 2031905.
  3. ^ Ancochea, Germán (1953). "Zeros of self-inversive polynomials". Proceedings of the American Mathematical Society. 4 (6): 900–902. doi:10.1090/s0002-9939-1953-0058748-8. ISSN 0002-9939. JSTOR 2031826.
  4. ^ Schinzel, A. (2005-03-01). "Self-Inversive Polynomials with All Zeros on the Unit Circle". The Ramanujan Journal. 9 (1–2): 19–23. doi:10.1007/s11139-005-0821-9. ISSN 1382-4090.
  5. ^ Vieira, R. S. (2017). "On the number of roots of self-inversive polynomials on the complex unit circle". The Ramanujan Journal. 42 (2): 363–369. arXiv:1504.00615. doi:10.1007/s11139-016-9804-2. ISSN 1382-4090.
  6. ^ Marden, Morris (1970). Geometry of polynomials (revised edition). Mathematical Surveys and Monographs (Book 3) United States of America: American Mathematical Society. ISBN 978-0821815038.{{cite book}}: CS1 maint: location (link)