Univariate polynomials and the contractibility of certain sets

Abstract

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 which are of opposite signs, the subset is contractible. For even $d$ and when $Q_d$ has two positive (resp. two negative) roots, the subset is contractible or empty. It is empty exactly when the constant term is positive, among the other even coefficients there is at least one which is negative, and all odd coefficients are positive (resp. negative).