# Reciprocal polynomial

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Reciprocal polynomial

In mathematics, for a polynomial "p" with complex coefficients,:$p\left(z\right) = a_0 + a_1z + a_2z^2 + ldots + a_nz^n ,!$we define the reciprocal polynomial, p*:

where $overline\left\{a\right\}_i$ denotes the complex conjugate of $a_i ,!$.

A polynomial is called self-reciprocal if $p\left(z\right) equiv p^\left\{*\right\}\left(z\right)$.

If the coefficients "a""i" are real then this reduces to "a""i" = "a""n"−"i". In this case "p" is also called a palindromic polynomial.

If "p"("z") is the minimal polynomial of "z"0 with |"z"0| = 1, and "p"("z") has real coefficients, then "p"("z") is self-reciprocal. This follows because

:.

So "z"0 is a root of the polynomial which has degree "n". But, the minimal polynomial is unique, hence :

A consequence is that the cyclotomic polynomials $Phi_n$ are self-reciprocal for $n > 1$; this is used in the special number field sieve to allow numbers of the form $x^\left\{11\right\} pm 1$, $x^\left\{13\right\} pm 1$, $x^\left\{15\right\} pm 1$ and $x^\left\{21\right\} pm 1$ to be factored taking advantage of the algebraic factors by using polynomials of degree 5, 6, 4 and 6 respectively - note that $phi$ of the exponents are 10, 12, 8 and 12.

* [http://mathworld.wolfram.com/ReciprocalPolynomial.html Reciprocal Polynomial] (on MathWorld)

References

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