Editing Reciprocal polynomial (section)

==Conjugate reciprocal polynomials{{anchor|Conjugate}}==

A polynomial is '''conjugate reciprocal''' if <math>p(x) \equiv p^{\dagger}(x)</math> and '''self-inversive''' if <math>p(x) = \omega p^{\dagger}(x)</math> for a scale factor {{math|''ω''}} on the [[unit circle]].<ref name=SV08>{{cite book | last1=Sinclair | first1=Christopher D. | last2=Vaaler | first2=Jeffrey D. | chapter=Self-inversive polynomials with all zeros on the unit circle | zbl=1334.11017 | editor1-last=McKee | editor1-first=James | editor2-last=Smyth | editor2-first=C. J. | title=Number theory and polynomials. Proceedings of the workshop, Bristol, UK, April 3–7, 2006 | location=Cambridge | publisher=[[Cambridge University Press]] | isbn=978-0-521-71467-9 | series=London Mathematical Society Lecture Note Series | volume=352 | pages=312–321 | year=2008 }}</ref>

If {{math|''p''(''z'')}} is the [[Minimal polynomial (field theory)|minimal polynomial]] of {{math|''z''<sub>0</sub>}} with {{math|1={{abs|''z''<sub>0</sub>}} = 1, ''z''<sub>0</sub> ≠ 1}}, and {{math|''p''(''z'')}} has [[real number|real]] coefficients, then {{math|''p''(''z'')}} is self-reciprocal.  This follows because

:<math>z_0^n\overline{p(1/\bar{z_0})} = z_0^n\overline{p(z_0)} = z_0^n\bar{0} = 0.</math>

So {{math|''z''<sub>0</sub>}} is a root of the polynomial <math>z^n\overline{p(\bar{z}^{-1})}</math> which has degree {{math|''n''}}.  But, the minimal polynomial is unique, hence 
:<math>cp(z) = z^n\overline{p(\bar{z}^{-1})}</math>
for some constant {{math|''c''}}, i.e. <math>ca_i=\overline{a_{n-i}}=a_{n-i}</math>. Sum from {{math|1=''i'' = 0}} to {{math|''n''}} and note that 1 is not a root of {{math|''p''}}. We conclude that {{math|1=''c'' = 1}}.

A consequence is that the [[cyclotomic polynomial]]s {{math|Φ<sub>''n''</sub>}} are self-reciprocal for {{math|''n'' > 1}}. This is used in the [[special number field sieve]] to allow numbers of the form {{math|''x''<sup>11</sup> ± 1, ''x''<sup>13</sup> ± 1, ''x''<sup>15</sup> ± 1}} and {{math|''x''<sup>21</sup> ± 1}} to be factored taking advantage of the algebraic factors by using polynomials of degree 5, 6, 4 and 6 respectively – note that {{math|''φ''}} ([[Euler's totient function]]) of the exponents are 10, 12, 8 and 12.{{Citation needed|date=November 2021}}

Per [[Cohn's theorem]], a self-inversive polynomial has as many roots in the [[unit disk]] <math>\{z\in\mathbb{C}: |z| < 1\}</math> as the reciprocal polynomial of its [[derivative]].<ref>{{Cite journal|last=Ancochea|first=Germán|date=1953|title=Zeros of self-inversive polynomials|url=https://www.ams.org/proc/1953-004-06/S0002-9939-1953-0058748-8/|journal=Proceedings of the American Mathematical Society|language=en|volume=4|issue=6|pages=900–902|doi=10.1090/S0002-9939-1953-0058748-8|issn=0002-9939|doi-access=free}}</ref><ref>{{Cite journal|last1=Bonsall|first1=F. F.|last2=Marden|first2=Morris|date=1952|title=Zeros of self-inversive polynomials|url=https://www.ams.org/proc/1952-003-03/S0002-9939-1952-0047828-8/|journal=Proceedings of the American Mathematical Society|language=en|volume=3|issue=3|pages=471–475|doi=10.1090/S0002-9939-1952-0047828-8|issn=0002-9939|doi-access=free}}</ref>