there exsit a real coefficient polynomial $g(x)$ of degree $n$. and for complex numbers $z(|z|=1)$ have $|g(z)|^2=1+|f(z)|^2$

Question

Let $n$ be give postive integers. such for any real coefficient polynomial $f(x)$ of degree $n$,show that:there exsit a real coefficient polynomial $g(x)$ of degree $n$. and for complex numbers $z(|z|=1)$ have $$|g(z)|^2=1+|f(z)|^2$$

maybe can use Rouche's theorem solve it?

Answer 1

The expression $$ 1+f(z)f(z^{-1}) $$ is symmetric under the map $z\mapsto z^{-1}$. Its roots come thus either in pairs $r,r^{-1}$ for real roots or as quadruples $ζ,\bar ζ,ζ^{-1},\bar ζ^{-1}$. Assign half the roots to $g$, the complex ones in conjugate pairs, and compute the product of linear factors. Add a constant factor $c$ so that $g(1)^2=1+f(1)^2$. $$ g(z)=c\prod(z-r_i)\prod(z-ζ_k)(z-\bar ζ_k) $$ is then one possible solution.