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- W3123374957 abstract "We consider the set $Pi ^*_d$ of monic polynomials $Q_d=x^d+sum _{j=0}^{d-1}a_jx^j$, $xin mathbb{R}$, $a_jin mathbb{R}^*$, 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." @default.
- W3123374957 created "2021-02-01" @default.
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- W3123374957 date "2021-01-21" @default.
- W3123374957 modified "2023-09-27" @default.
- W3123374957 title "Univariate polynomials and the contractibility of certain sets" @default.
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