Polynomial with all zeros on a circle and many real coefficients On a circle (or a line) $\Omega$ in the complex plane that is not symmetric w.r.t. the real axis, choose $n\ge5$ distinct points $z_1,...,z_n$ and consider the polynomial $p(z)=\prod_j(z-z_j)=z^n+a_{n-1}z^{n-1}+\cdots+a_0$. Question: How many of the $a_k$'s can be reals at most ?
Intuitively, imagining something like a Java applet, there are "more or less" $n-1$ degrees of freedom to move the points around on $\Omega$, so we might reasonably expect that it is possible to choose them (say for an appropriate $\Omega$) such that all but one $a_{k_0}$ are reals.


*

*How could that be rigorously proven? 

*If it is true, can it be done for any $k_0\in\lbrace0,...,n-1\rbrace$? 

*How to find a concrete solution? 

*Can such a polynomial even be rational, i.e. such that $a_{k_0}\in\mathbb Q[i]$ and $a_k\in\mathbb Q$ for $k\ne k_0$?

 A: Suppose it is a line, passing through the origin under the angle $\phi$.
Then your polynomial must be
$$z^n+a_1z^{n-1}+...+a_0=\prod_{j=1}^n(z-t_je^{i\phi}),$$
where $t_j$ are real. Vjeta's formulas give
$$a_k=\pm\sum t_{i_1}...t_{i_k} e^{ik\phi},$$
Now how can $a_k$ be real?
First way: $e^{ik\phi}$ is real. For how many $k=1...n$ this can happen, is easy to find out.
Second way: $$b_k:=\sum t_{i_1}...t_{i_k}=0.$$
Of course this can happen for all $k$ if all $t_k=0$. 
If you want to exclude $t_j=0$ than the question is reduced to
"how many zero coefficients can have a polynomial with all roots real and non-zero ?". I mean the real
polynomial $\prod(z-t_k)$, whose coefficients are $\pm b_k$.
For this real polynomial, you can use the following theorem of Descartes:
The number of positive zeros of a real polynomial is at most the number of sign changes
in the sequence of coefficients (which is at most the number of non-zero coefficients minus 1).
Same applies to the number of negative zeros if you make the change of the variable $x\to-x$,
which changes the sign switches but does not change the number of non-zero coefficients.
If you want all roots to be distinct, at most one of them is zero.
I leave the details to you.
