  We consider real univariate polynomials with all roots real. Such a
polynomial with $c$ sign changes and $p$ sign preservations in the sequence of
its coefficients has $c$ positive and $p$ negative roots counted with
multiplicity. Suppose that all moduli of roots are distinct; we consider them
as ordered on the positive half-axis. We ask the question: If the positions of
the sign changes are known, what can the positions of the moduli of negative
roots be? We prove several new results which show how far from trivial the
answer to this question is.
