Algebraic Moments – Real Root Finding and Related Topics –

Polynomial root finding is a classical and well studied problem in mathematics with practical applications in many areas. Various relevant questions can be reformulated into the task of finding all common roots of a given system of polynomials in several variables, the so-called algebraic variety or zero set. Traditionally, root finding algorithms are implemented in exact arithmetics over algebraically closed fields and fall into the area of computational algebraic geometry. However, in practical applications data is often uncertain and only known up to a certain finite precision. Algorithms in the new and promising area of numerical polynomial algebra or numerical algebraic geometry reflect this issue. Just like in linear algebra, the use of numerical methods has the potential of reducing the complexity and storage requirements and thus allows to treat problems of larger size. Almost all existing numerical algebraic methods perform computations over the field of complex numbers, while in practical problems the variables are mostly real valued. In this thesis we address the need for numerical software in real algebraic geometry by proposing two algorithms for computing real roots or even roots in certain semialgebraic sets. A novel semi-definite representation of the real radical ideal allows a unified treatment for the computation of real and complex roots as well as the construction of border and Gröbner bases of the ideal or its real radical ideal respectively. We address the issue of augmenting existing symbolic-numeric algorithms with real algebraic features and provide a first step towards efficient algebraic algorithms for computations over the real numbers. After recalling some basic concepts from commutative algebra and algebraic geometry, two existing root finding methods are discussed. Nonlinear parametric programming is introduced as an example, which can be tackled using polynomial algebra for presolving. The precomputation, based on solving first order optimality conditions, is particularly useful in time critical applications of parametric programming. Model predictive control, where an optimal control problem with changing initial state is solved in real time, demonstrates an important example. The first approach uses symbolic computation and proceeds by precomputing Gröbner basis and so-called generalized companion matrices. The online algorithm reduces