Reciprocal polynomial

In mathematics, for a polynomial p with complex coefficients,

$p(z)=a_{0}+a_{1}z+a_{2}z^{2}+\ldots +a_{n}z^{n}\,\!$

we define the reciprocal polynomial, p*

$p^{*}(z)={\overline {a}}_{n}+{\overline {a}}_{n-1}z+\ldots +{\overline {a}}_{0}z^{n}=z^{n}{\overline {p({\bar {z}}^{-1})}}$

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 z₀ with |z₀| = 1, and p(z) has real coefficients, then p(z) is self-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 $z^{n}{\overline {p({\bar {z}}^{-1})}}$ which has degree n. But, the minimal polynomial is unique, hence

$p(z)=z^{n}{\overline {p({\bar {z}}^{-1})}}.$

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

• Reciprocal Polynomial (on MathWorld)