  We study the stratification of the space of monic polynomials with real
coefficients according to the number and multiplicities of real zeros. In the
first part, for each of these strata we provide a purely combinatorial chain
complex calculating (co)homology of its one-point compactification and describe
the homotopy type by order complexes of a class of posets of compositions. In
the second part, we determine the homotopy type of the one-point
compactification of the space of monic polynomials of fixed degree which have
only real roots (i.e., hyperbolic polynomials) and at least one root is of
multiplicity $k$. More generally, we describe the homotopy type of the
one-point compactification of strata in the boundary of the set of hyperbolic
polynomials, that are defined via certain restrictions on root multiplicities,
by order complexes of posets of compositions. In general, the methods are
combinatorial and the topological problems are mostly reduced to the study of
partially ordered sets.
