Combinatorics and semi-algebraic geometry of orbit spaces: Hyperbolic and stable polynomials, the degree principle and few inequalities defining orbit spaces

Abstract

<p>Algebraic geometry studies the set of common zeros of a system of polynomials in one or several variables - most commonly over algebraically closed fields. Real algebraic geometry studies sets defined by a finite system of polynomial inequalities. <p>A starting point for modern real algebraic geometry can be traced back to Hilbert: In 1888, Hilbert showed the existence of nonnegative polynomials which are not sums of squares of polynomials. In 1900, he posed his famous 23 problems and, in particular, the 17th can be stated as follows: Is every nonnegative polynomial a sum of squares of rational functions? <p>Artin's solution to Hilbert’s 17th problem can be seen as a kick-off for real algebraic geometry. This thesis deals with real symmetric polynomials and the results are closely related to an answer of Hilbert's 17th problem for symmetric polynomials: A characterization of all symmetric nonnegative polynomials.