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In the mathematical area of [[algebra]], given a [[polynomial]] ''p'' with coefficients from an arbitrary [[Field (mathematics)|field]] such as:
:<math>p(x) = a_0 + a_1x + a_2x^2 + \ldots + a_nx^n, \,\!</math>
we define the '''reciprocal polynomial''', '''p*'''by:<ref>{{harvnb|Roman|1995|loc=pg.37}}</ref>
:<math>p^*(x) = a_n + a_{n-1}x + \ldots + a_0x^n = x^n p(x^{-1}).</math>
Essentially, the coefficients are written in reverse order.


In the special case that the [[polynomial]] ''p'' has [[complex number|complex]] coefficients, that is,
:<math>p(z) = a_0 + a_1z + a_2z^2 + \ldots + a_nz^n, \,\!</math>
the ''conjugate reciprocal polynomial'', '''p*''' given by,
:<math>p^*(z) = \overline{a_n} + \overline{a_{n-1}}z + \ldots + \overline{a_0}z^n = z^n\overline{p(\bar{z}^{-1})},</math>


where <math>\overline{a_i}</math> denotes the [[complex conjugate]] of <math>a_i \,\!</math>, is called the reciprocal polynomial when no confusion can arise.
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A polynomial is called '''self-reciprocal''' if <math>p(x) \equiv p^{*}(x)</math>.
 
The coefficients of a self-reciprocal polynomial satisfy ''a''<sub>''i''</sub> = ''a''<sub>''n''−''i''</sub>, and in this case ''p'' is also called a [[palindromic polynomial]]. In the conjugate reciprocal case, the coefficients must be [[Real number|real]] to satisfy the condition.
 
== Properties ==
Reciprocal polynomials have several connections with their original polynomials, including:
# α is a root of polynomial ''p'' if and only if α<sup>−1</sup> is a root of ''p''*.<ref>{{harvnb|Pless|1990|loc=pg. 57}}</ref>
# If ''p''(''x'') ≠ ''x'' then ''p'' is [[Irreducible polynomial|irreducible]] if and only if ''p''* is irreducible.<ref>{{harvnb|Roman|1995|loc= pg. 37}}</ref>
# ''p'' is [[Primitive polynomial (field theory)|primitive]] if and only if ''p''* is primitive.<ref>{{harvnb|Pless|1990|loc=pg. 57}}</ref>
 
Other properties of reciprocal polynomials may be obtained, for instance:
* If a polynomial is self-reciprocal and irreducible then it must have even degree.<ref>{{harvnb|Roman|1995|loc= pg. 37}}</ref>
 
==Properties of conjugate reciprocal polynomials==
 
If ''p''(''z'') is the [[Minimal polynomial (field theory)|minimal polynomial]] of ''z''<sub>0</sub> with |''z''<sub>0</sub>| = 1, <math>z_0\neq1</math>, and ''p''(''z'') has [[real number|real]] coefficients, then ''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 ''z''<sub>0</sub> is a root of the polynomial <math>z^n\overline{p(\bar{z}^{-1})}</math> which has degree ''n''. But, the minimal polynomial is unique, hence
:<math>cp(z) = z^n\overline{p(\bar{z}^{-1})}</math>
for some constant ''c'', i.e. <math>ca_i=\overline{a_{n-i}}=a_{n-i}</math>. Sum from ''i'' = 0 to ''n'' and note that 1 is not a root of ''p''. We conclude that ''c'' = 1.
 
A consequence is that the [[cyclotomic polynomial]]s <math>\Phi_n</math> are self-reciprocal for <math>n > 1</math>; this is used in the [[special number field sieve]] to allow numbers of the form <math>x^{11} \pm 1</math>, <math>x^{13} \pm 1</math>, <math>x^{15} \pm 1</math> and <math>x^{21} \pm 1</math> to be factored taking advantage of the algebraic factors by using polynomials of degree 5, 6, 4 and 6 respectively - note that <math>\phi</math> ([[Euler's totient function]]) of the exponents are 10, 12, 8 and 12.
 
==Application in coding theory==
 
The reciprocal polynomial finds a use in the theory of [[Cyclic code|cyclic error correcting codes]]. Suppose ''x''<sup>''n''</sup> - 1 can be factored into the product of two polynomials, say ''x''<sup>''n''</sup> - 1 = ''g''(''x'')''p''(''x''). When ''g''(''x'') generates a cyclic code ''C'', then the reciprocal polynomial ''p''*(x) generates ''C''<sup>⊥</sup>, the [[orthogonal complement]] of ''C''.<ref>{{harvnb|Pless|1990|loc = pg. 75, Theorem 48}}</ref>
Also, ''C'' is ''self-orthogonal'' (that is, ''C'' ⊆ ''C''<sup>⊥</sup>), if and only if  ''p''*(''x'') divides ''g''(''x'').<ref>{{harvnb|Pless|1990|loc = pg. 77, Theorem 51}}</ref>
 
==See also==
*[[Schur Transform]]
 
== Notes ==
{{reflist}}
 
==References==
{{Refimprove|date=June 2008}}
* {{citation|first=Vera|last=Pless|title=Introduction to the Theory of Error Correcting Codes|edition=2nd|publisher=Wiley-Interscience|place=New York|year=1990|isbn=0-471-61884-5}}
 
* {{citation|first=Steven|last=Roman|title=Field Theory|publisher=Springer-Verlag|place=New York|year=1995|isbn=0-387-94408-7}}
 
== External links ==
* [http://mathworld.wolfram.com/ReciprocalPolynomial.html Reciprocal Polynomial] (on [[MathWorld]])
 
[[Category:Polynomials]]

Revision as of 13:15, 20 February 2014


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