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 for degree less or equal to 5), one fixes the pairs (compatible with Rolle's theorem) of numbers of positive and negative roots of the polynomial and its non-constant derivatives. A third problem concerns polynomials with all roots real. One considers the sign pattern and the order in which the moduli of its positive and negative roots are arranged on the positive half-line. There are examples of pairs (sign pattern, order of moduli) compatible with Descartes' rule of signs that are not realizable. And there are various questions about the discriminant of the general family of uni-variate polynomials. The non-trivial answers to these simply formulated problems will give students and scholars a better understanding of uni-variate polynomials.

Autorentext

Vladimir Petrov Kostov is born in 1959 in Bulgaria. He has defended his PhD thesis at the Faculty of Mechanics and Mathematics of Moscow State University in 1990 and has been on a postdoc position at the University of Utrecht in 1990-1991. Since 1991 he is working at the Laboratory of Mathematics of the University of Nice, France. His research fields are:

1. the analytic theory of systems of linear differential equations, the Riemann-Hilbert and the Deligne-Simpson problem (see [1]);
2. uni-variate and in particular hyperbolic polynomials (i.e. polynomials with all roots real, see [2-6]); this includes also realization problems about uni-variate polynomials in the context of Descartes' rule of signs;
3. analytic properties of the partial theta function (see [6-8]).
   [1] V.P. Kostov, The Deligne-Simpson problem -- a survey. Journal of Algebra 281 No. 1 (2004) 83 -108.
   [2] V.P. Kostov, Topics on hyperbolic polynomials in one variable. Panoramas et Synthèses 33 (2011), vi 141 p. SMF.
   [3] J. Forsgård, V.P. Kostov and B.Z. Shapiro, Could René Descartes have known this?, Experimental Mathematics vol. 24, issue 4 (2015), 438-448, DOI: 10.1080/10586458.2015.1030051.
   [4] V.P. Kostov, On realizability of sign patterns by real polynomials, Czechoslovak Math. J. 68 (143) (2018), no. 3, 853-874.
   [5] V.P. Kostov, Hyperbolic polynomials and rigid orders of moduli, Publicationes Mathematicae Debrecen100/1-2 (2022), 119-128 DOI: 10.5486/PMD.2022.9068.
   [6] V.P. Kostov and B.Z. Shapiro, Hardy-Petrovitch-Hutchinson's problem and partial theta function, Duke Math. J. 162, No. 5 (2013) 825-861.
   [7] V.P. Kostov, On the zeros of a partial theta function, Bull. Sci. Math. 137, No. 8 (2013) 1018-1030.
   [8] V.P. Kostov, On the double zeros of a partial theta function, Bull. Sci. Math. 140, No. 4 (2016) 98-111.