  The classical Descartes' rule of signs limits the number of positive roots of
a real polynomial in one variable by the number of sign changes in the sequence
of its coefficients. One can ask the question which pairs of nonnegative
integers $(p,n)$, chosen in accordance with this rule and with some other
natural conditions, can be the pairs of numbers of positive and negative roots
of a real polynomial with prescribed signs of the coefficients. The paper
solves this problem for degree $8$ polynomials.
