Descartes's rule of signs
(aka, Descartes' sign rule)

Given a real polynomial, the number of sign changes in the
coefficients (ignoring ones that don't occur) gives you the maximum
number of -positive- roots, and indeed the actual number of positive
roots agrees with this mod $2$, if counted with multiplicities.

You can likewise get the maximum number of negative roots by applying
this to $p(-x)$.

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You can't improve Descartes' theorem in the positive case:
given a number of sign changes, any allowed number of positive roots
can occur; in particular, the max number can occur.
(Q: can the powers of these be also specified arbitrarily,
 IE, where the sign changes occur?)

OTOH, you can't achieve all possible combinations of
(positive and negative) roots so allowed.
You can always achieve the max number of positives and negatives,
but there are -some- restrictions.
(see the Monthly, 1998, 1999, 2004)
What the possibilities are is, AFAICT, not known.

Nor, frankly, do I much care. This is a barely cute problem,
based on what I know (maybe it has hidden depth...).
