Reciprocal Polynomial

In mathematics, for a polynomial p with complex coefficients,
p(z) = a_0 + a_1z + a_2z^2 + \ldots + a_nz^n
we define
p^*(z) = \overline{a_n} + \overline{a_{n-1}}z + \ldots + \overline{a_0}z^n = z^n\overline{p(1/\bar{z})}
where \overline{a_i} denotes the complex conjugate of a_i A polynomial is called reciprocal if p(z) = p*(z). If the coefficients ai are real then this reduces to ai = an-i. If p(z) is the minimal polynomial of z0 with |z0| = 1, and p(z) has real coefficients, then p(z) is reciprocal. This follows because,
z_0^n\overline{p(1/\bar{z_0})} = z_0^n\overline{p(z_0)} = z_0^n\bar{0} = 0.
So z0 is a root of the polynomial z^n\overline{p(1/\bar{z})} which has degree n. But, the minimal polynomial is unique, hence
p(z) = z^n\overline{p(1/\bar{z})}.
A consequence is that the cyclotomic polynomials \Phi_n are reciprocal for n > 1.

 

<< PreviousWord BrowserNext >>
andregota galndez
electrical wiring
borjigin
uss republic (ncc 1371)
iwakura mission
wcnn (am)
golden virgins
john peel (disambiguation)
united states embargo against cuba
agia eirini (kefalonia), greece
anishinaabe
ta'qali stadium
punk magazine
electrical wiring (uk)
augustana college (south dakota)
the call of ktulu
high spirits
thomas stephens
keweenaw waterway
sinderhope
turf toe
mary eleanor wilkins freeman
day of radiance
nepal workers and peasants party
four great sf novels
anna vyrubova
ermenguer
linear no threshold model
kajillion
chinawhite
nepal revolutionary students union
dippet
hsh
communist mazdoor kissan party
karak district
sbk records
chew valley lake
duban
radiation hormesis
somerset hamilton butler, 1st earl of carrick
list of south carolina state parks
ashuwillticook rail trail
raduga kh 15
u.s. senate election, 1990