Max modulus Constrained Optimization

Question

Let $z_1,z_2\ldots,z_n\in\mathbb{C}$ such that $$\sum_{i=1}^n{z_i^j}=1\quad,\:\forall j=1,2,\ldots,n-1$$ and $$\sum_{i=1}^{n}z_i^n=q\qquad\textrm{ with } |q|<1.$$ I am interested in finding a bound on the maximum modulus $$\max_{1\leq i\leq n}\{|z_i|\}$$ Specifically, if conditions can be imposed on $|q|$ such that these bounds are less that 1 i.e. all $z_i$ lie within the unit circle.

Answer 1

Reformulating, we want some conditions on $q$ such that all the roots of the polynomial $z^n-z^{n-1}+(1-q)$ lie inside the unit circle. This is pretty much the same exercise appearing at page 127 of my notes, serving as an introduction to the original proof of the Fundamental Theorem of Algebra provided by Gauss. Let $\tau=1-q$. Given the relation between the topological degree of curves and the number of roots of polynomials inside disks, it is enough to understand which values of $\tau$ ensure that the origin belongs to the inner loop of the parametric curve $$\gamma(s)=\left(\cos(ns)-\cos((n-1)s)+\tau,\sin(ns)-\sin((n-1)s)\right).$$ For instance, here it is our curve for $n=7$ and $\tau=0$:

The upper part of the inner loop is given by values of $s$ between $0$ and $\frac{\pi}{2n+1}$. If we want the origin to be enclosed by the inner loop, we need that $0<\tau<2\sin\left(\frac{\pi}{4n+2}\right)$. If we allow complex $\tau$s,

$$ \left\|\tau-\sin\left(\tfrac{\pi}{4n+2}\right)\right\|\leq \tfrac{1}{2}\,\sin\left(\tfrac{\pi}{4n+2}\right)$$ is a sufficient condition for $x^{n}-x^{n-1}+\tau$ to have all its zeroes inside the unit disk.