Reciprocal polynomial


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 the reciprocal polynomial, p*:p^*(z) = overline{a}_n + overline{a}_{n-1}z + ldots + overline{a}_0z^n = z^noverline{p(ar{z}^{-1})}

where overline{a}_i denotes the complex conjugate of a_i ,!.

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

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

:z_0^noverline{p(1/ar{z_0})} = z_0^noverline{p(z_0)} = z_0^nar{0} = 0.

So "z"0 is a root of the polynomial z^noverline{p(ar{z}^{-1})} which has degree "n". But, the minimal polynomial is unique, hence :p(z) = z^noverline{p(ar{z}^{-1})}.

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^{11} pm 1, x^{13} pm 1, x^{15} pm 1 and x^{21} 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.

See also: Schur Transform

External links

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

References


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