Recent posts on polynomials have got me thinking.
I want to find the real roots of a polynomial with real coefficients in one real variable $x$. I know I can use a Sturm Sequence to find the number of roots between two chosen limits $a < x < b$.
Given that $p(x) = \sum_{r=0}^n a_rx^r$ with $a_n = 1$ what are the tightest values for $a$ and $b$ which are simply expressed in terms of the coefficients $a_r$ and which make sure I capture all the real roots?
I can quite easily get some loose bounds and crank up the computer to do the rest, and if I approximate solutions by some algorithm I can get tighter. But I want to be greedy and get max value for min work.