math lessons - 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

$p^*(z) = \overline{a_n} + \overline{a_{n-1}}z + \ldots + \overline{a_0}z^n = z^n\overline{p(1/\bar{z})}$

where $\overline{a_i}$ denotes the complex conjugate of $a i$

A polynomial is called reciprocal if p(z) = p^*(z).

If the coefficients a_i are real then this reduces to a_i = a_n-i.

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

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

A consequence is that the cyclotomic polynomials $Φ n$ are reciprocal for $n > 1$ .

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