On a problem inspired by Descartes' rule of signs

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.

Keywords

• real polynomial in one variable
• hyperbolic polynomial
• sign pattern
• descartes' rule of signs