Vladimir Kostov: Hyperbolic polynomials, moduli of their roots and Descartes' rule of signs

Abstract.

A real univariate polynomial is hyperbolic if all its roots are real. Descartes' rule of signs applied to hyperbolic polynomials with no vanishing coefficients says that such a degree $d$ polynomial has exactly $c$ positive and $p$ negative roots, $c+p=d$, where $c$ and $p$ are the numbers of sign changes and sign preservations in the sequence of $d+1$ coefficients of the polynomial. One can consider the chain of $d$ moduli of roots (we assume that they are distinct) of a given polynomial on the positive half-axis. One can try to find a relationship between the positions in this chain of the moduli of the positive/negative roots and the positions of the sign changes/preservations in the sequence of coefficients. We focus especially on the case when the order of the moduli determines the sequence of coefficients (such an order is called em rigid) and the case when the sequence of the signs of the coefficients (such a sequence is called canonical) determines the order of the moduli on the positive half-axis.