The disconnectedness of certain sets defined after uni-variate polynomials

Abstract

We consider the set of monic real uni-variate polynomials of a given degree $d$ with non-vanishing coefficients, with given signs of the coefficients and with given quantities $pos$ of their positive and $neg$ of their negative roots (all roots are distinct). For $d\geq 6$ and for signs of the coefficients $(+,-,+,+,\ldots ,+,+,-,+)$, we prove that the set of such polynomials having two positive, $d-4$ negative and two complex conjugate roots, is not connected. For $pos+neg\leq 3$ and for any $d$, we give the exhaustive answer to the question for which signs of the coefficients there exist polynomials with such values of $pos$ and $neg$.