Take the 2-minute tour ×  This exrcise is from the first section
of Marden:  Exercise 12. Let the interior of a piecewise regular
curve $C$ contain the origin $\cal O$ and be star-shaped with
respect to $\cal O$. If the complex numbers $a_1, a_2, \ldots,
a_m$ are given, then $n$ and $b_{m+1}, b_{m+2}, \ldots, b_n$ may
be determined so that all of the zeros of  $F(z) = 1 + a_1 z +
\cdots + a_m z^m + b_{m+1} z^{m+1} + \cdots + b_n z^n$  lie on
$C$.  Hint: Choose the zeros $\zeta_j$ of $G(z) = z^n F(1/z)$ so
that $z_j = 1/\zeta_j$ are points of $C$ and so that the Newton-
Girard formulas  $s_k + s_{k-1} a_1 + \cdots + k a_k = 0$  are
satisfied by the sums $s_r$ of the $r^{\text{th}}$ powers of the
$\zeta_j$.  I understand how the requirements of the hint will
imply the result, but I do not know how to establish them.
Indeed, if we can satisfy those requirements, then the remaining
$b_j$ will be determined by the further Newton-Girard formulas.
This is a generalization of the previous problem in the book:
Exerci