Uni-variate Polynomials in Analysis: Hyperbolic Polynomials, Order of Moduli, and Descartes' Rule of Signs

The book summarizes recent results on problems with uni-variate polynomials. The first of them reads: given the signs of the coefficients of a real polynomial (i. e. its sign pattern), for which pairs of prescribed numbers of positive and negative roots (compatible with Descartes’ rule of signs) can one find such a polynomial? For each degree greater or equal to 4, there are non-realizable cases. The problem is resolved for degree less or equal to 8. In another realization problem (resolved...