On a problem inspired by Descartes' rule of signs (2404.13943v1)

Abstract: We study real univariate polynomials with non-zero coefficients and with all roots real, out of which exactly two positive. The sequence of coefficients of such a polynomial begins with $m$ positive coefficients followed by $n$ negative followed by $q$ positive coefficients. We consider the sequence of moduli of their roots on the positive real half-axis; all moduli are supposed distinct. We mark in this sequence the positions of the moduli of the two positive roots. For $m=n=2$, $n=q=2$ and $m=q=2$, we give the exhaustive answer to the question which the positions of the two moduli of positive roots can be.