[Cross-posted from Math.SE because I got no responses there.]
Given a polynomial in $e^{\mathrm{i}k t}$ of the form $$ p(t) = \sum_{-n\leq k\leq n} c_k e^{\mathrm{i}k t} $$ with $\bar c_{-k} = c_k$, is there a good way of characterising how close its roots can be in terms of its coefficients?
Clearly this depends on coefficients, for example $1-\epsilon-\cos x$ has roots $\sqrt{\epsilon}$ apart, but for general coefficients, is there any good result? I am secretly hoping for minimum distance of something like $A n^{-1}$ (which is the case for $\sin n x$) for non-almost-degenerate coefficients.
If you know a result for any kind of orthogonal polynomials (like Chebyshev polynomials), that would also be very helpful. Also, I know there are some results for asymptotic distributions of eigenvalues of Toeplitz matrices, but I am interested in a result for a fixed polynomial $p$ and its coefficients, and not just $n\to\infty$.
Edit. I am mainly interested in an algorithm that, given a polynomial as above, computes the shortest gap between two adjacent roots of the polynomial.
The motivation is as follows. The simplest method for calculating a root of a function numerically is the bisection method, which only works so long as you know an interval that brackets a root (an interval on which the function changes sign). In general, calculating all roots of a black-box function is really hard. So without going to more involved algorithms, the simplest algorithm to find all (simple) roots must be this one: find the shortest distance $\delta$ between adjacent roots, sample the function in intervals of length $\delta$, find the root in each interval with a sign change in it. Of course, this doesn't work without knowing $\delta$. This is not a brilliant algorithm, there are better ones out there, but I am interested in finding out how I could make it work.
So the question is: given a trigonometric polynomial with only simple roots, how can I calculate the shortest distance between adjacent roots, so that the above scheme would succeed?
Example. If $t_{1,2}$ are two roots, I can write $$ 2\max_{[t_1,t_2]}|f(t)| \leq \int_{t_1}^{t_2}|f'(t)|\,dt \leq |t_2-t_1|\max_{[t_1,t_2]}|f'(t)|. $$ So at least $$ |t_1-t_2| \geq 2\frac{\max_I |f|}{\max_I |f'|}. $$ Then by Bernstein's inequality, $$ \max_I |f'| \leq \|f'\|_\infty \leq n \|f\|_\infty, $$ so $$ |t_1-t_2| \geq \frac2m \frac{\max_I|f|}{\max_{[-\pi,\pi]}|f|}. $$ But then I don't know quite how to deal with the right-hand side.
