  We consider the set $\Pi ^*_d$ of monic polynomials $Q_d=x^d+\sum
_{j=0}^{d-1}a_jx^j$, $x\in \mathbb{R}$, $a_j\in \mathbb{R}^*$, having $d$
distinct real roots, and its subsets defined by fixing the signs of the
coefficients $a_j$. We show that for every choice of these signs, the
corresponding subset is non-empty and contractible. A similar result holds true
in the cases of polynomials $Q_d$ of even degree $d$ and having no real roots
or of odd degree and having exactly one real root. For even $d$ and when $Q_d$
has exactly two real roots, the subset is contractible or empty. It is empty
exactly when the constant term and all odd coefficients are positive and there
is at least one even coefficient which is negative.
