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*:

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

:.

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 $x^\{11\}\; pm\; 1$, $x^\{13\}\; pm\; 1$, $x^\{15\}\; pm\; 1$ 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