Positive polynomials and sums of squares

点击翻译按钮获取中文摘要

Real algebraic geometry studies geometrical objects arising in connection with the mathematical modeling of the "real world", i.e., the mathematical description of physical objects as well as of natural and man-made processes in science and technology. The most basic functions considered in real algebraic geometry are the polynomial functions in n real variables with real coefficients. The domains considered are subsets of real n-space defined by polynomial equations and inequalities. A basic problem is to determine the minimum (or maximum) value of such a function on such a domain. This is a hard problem in general, but various fast algorithms based on semidefinite programming can be used to obtain estimates for the minimum value. The theoretical basis for these algorithms is the close relationship that exists between positive polynomials and sums of squares. The study of the relationship between positive polynomials and sums of squares is an old subject, dating back to work of Hilbert in the late 19th century. Landmarks include Artin's solution of Hilbert's 17th problem in 1927, Krivine's Positivstellensatz in 1964, and Schmuedgen's Positivstellensatz (dealing with the case when the domain in question is compact) in 1991. The main object of the proposed research is to understand this relationship better, and apply this information to understand better how these fast algorithms will perform in practical situations.